Reflection loss is usually calculated and reported as a function of the thickness of microwave absorption material. However, misleading results are often obtained since the principles imbedded in the popular methods contradict the fundamental facts that electromagnetic waves cannot be reflected in a uniform material except when there is an interface and that there are important differences between the concepts of characteristic impedance and input impedance. In this paper, these inconsistencies have been analyzed theoretically and corrections provided. The problems with the calculations indicate a gap between the background knowledge of material scientists and microwave engineers and for that reason a concise review of transmission line theory is provided along with the mathematical background needed for a deeper understanding of the theory of reflection loss. The expressions of gradient, divergence, Laplacian, and curl operators in a general orthogonal coordinate system have been presented including the concept of reciprocal vectors. Gauss’s and Stokes’s theorems have been related to Green’s theorem in a novel way.

There is considerable research interest in ferromagnetic resonance1–3 and microwave absorption.4,5 However, there are also problems from contemporary popular measurements6–14 when reflection loss from microwave absorption material is characterized. Any parameter used in characterizing a material should take into account the intrinsic property of the material independent of the size of the material and the circuit of the measurement. However, in the conventional method, reflection loss is incorrectly reported as a function of sample thickness calculated with respect to the position within the measured material. It is known that the longer microwaves travel in a microwave absorbing material, the more waves will be absorbed. Thus, the value of reflection loss from the conventional calculations can be artificially adjusted by sample thickness. A more serious misleading problem is that the calculations5,15,16 give rise to the artificial result that at a certain frequency of the microwave and beyond a specific sample thickness, the thicker the sample, the less microwaves are absorbed.

Clearly not much theoretical knowledge is needed to identify the problems mentioned above. But the fact that the inaccurate calculations have already become generally accepted and that the apparent wrong results can be published are indications that there is a lack of theoretical understanding among material researchers. Thus, a theoretical discussion is included in Sections II–IV and subsequently the problems in the calculations are analyzed and corrected in Section V. The errors can be easily recognized because the calculations are inconsistent with the well-established truth that electromagnetic waves cannot be reflected within a single phased material unless an interface is present. It is demonstrated here that the problems are caused by confusing the characteristic impedance of the material with the input impedance of the circuit. It is proved theoretically (Eqs. 33, 44, and 49 below) that reflection loss for microwave absorption has the same meaning as the scattering parameter s11 which is called return loss in microwave engineering if it is expressed in decibel units (dB).17–19 The theoretical background is presented here succinctly and simply. The discussion considering the matching and mismatching of impedances in Eqs. 2124 provides an easy way for material scientists to understand transmission line theory. The mathematical problems involved in the calculations are presented in Section VI.

The mathematical background in the Appendices provide the necessary skills to carry out more extensive theoretical studies. The ultimate goal of science is to attain a general and abstract understanding instead of just an intuitive perception. The gradient, divergence, Laplacian, and curl operators between orthogonal coordinate systems are discussed in general terms in Subsection 2 of the  Appendix. The novel section in this  Appendix involves the inclusion of the concept of reciprocal vectors and the way in which Green’s theorem is connected with Gauss’s and Stokes’s theorems.

Microwave measurement is based on transmission line theory in which the response of the material to microwaves is modeled by the responses of the magnetic and electric dipoles to the variations of magnetic and electric fields from electromagnetic waves.20 Resistive absorption can be negated for insulating ferrites. Electromagnetic waves are transmitted by the oscillations of electric and magnetic dipoles in insulating ferrites. For a precise model, both the inductor and the capacitor in Box A of Fig. 1 should respectively be shunted with a resistor of infinite resistance. However, in practice both the resistors can be omitted because the resistance is infinite for insulating material. In such measurements, microwaves are represented by voltage V and current I the values of which are oscillated at microwave frequency and the material is modeled by a series of slices of width dx with distributive inductance LM and capacitance CM as shown in Box A of Fig. 1. For a slice of dx in the material, the voltage drops across the inductance by LMdx(∂I/∂t) and the current reduces by CMdx(∂V/∂t) since this small current goes through the capacitor. These changes are described by the following Telegrapher’s equations, Eqs. 1 and 2.

$∂V∂x=−LM∂I∂t$
(1)
$∂I∂x=−CM∂V∂t$
(2)
FIG. 1.

Ferrite as a microwave absorbing material. The figure in box A represents a modeled circuit for any successive slice of an insulating material with width dx. Distributive inductance LM and capacitance CM are given as values per unit length. ZM(d) is the input impedance of the absorbing material ZM(x) at x = d and Zl(0) is the impedance of the transmission line at x = 0. ZM and Zl are the characteristic impedances of the material and the transmission lines (or the measuring equipment), respectively. Zl is usually 50 Ω. ε and μ are the permittivity and permeability of the material, respectively. The values εr and μr are relative to the values ε0 and μ0 in vacuum, respectively. A metal-backed ferrite structure can be viewed as a ferrite medium terminated with a load with ZL(x = d) = 0.

FIG. 1.

Ferrite as a microwave absorbing material. The figure in box A represents a modeled circuit for any successive slice of an insulating material with width dx. Distributive inductance LM and capacitance CM are given as values per unit length. ZM(d) is the input impedance of the absorbing material ZM(x) at x = d and Zl(0) is the impedance of the transmission line at x = 0. ZM and Zl are the characteristic impedances of the material and the transmission lines (or the measuring equipment), respectively. Zl is usually 50 Ω. ε and μ are the permittivity and permeability of the material, respectively. The values εr and μr are relative to the values ε0 and μ0 in vacuum, respectively. A metal-backed ferrite structure can be viewed as a ferrite medium terminated with a load with ZL(x = d) = 0.

Close modal

In the metric system, the distributive inductance LM has units of henries/m and the distributive capacitance CM has units of farads/m. The propagating microwave obeys Eqs. 3 and 4 which are derived from Telegrapher’s equations.20

$∂2V(t−xυp)∂x2=LMCM∂2V(t−xυp)∂t2$
(3)
$∂2I(t−xυp)∂x2=LMCM∂2I(t−xυp)∂t2$
(4)

$υp$ is the speed of microwave propagating in the absorbing material along x in Fig. 1. Equation 5 shows that Eq. 3 is consistent with Eq. 4 if the functions have the form $eik(t−xυp)$.19

$1υp=LMCM=εμ=ε0μ0εrμr=1cεrμr$
(5)

c is the velocity of the microwave in vacuum. From Eqs. 1 and 5

$∂V(t−xυp)∂x=−1υpV(t−xυp)=−LM∂I(t−xυp)∂t$
(6)

What Eq. 6 means is that

$V(t−xυp)=V+(t,x)=LMυpI(t−xυp)=LMCMI+(t,x)=ZMI+(t,x)=μεI+(t,x)=μ0μrε0εrI+(t,x)$
(7)

Similarly

$V−(t,x)I−(t,x)=V(t+xυp)I(t+xυp)=V0−I0−=−LMυp=−LMCM=−ZM=−με=−μ0μrε0εr$
(8)

V+(t, x) and I+(t, x) represent the forward wave and V-(t, x) and I-(t, x) the reflected wave. The absorption property of the material is characterized by LM and CM which in turn are related to the relative permittivity εr and permeability μr of the material.19 The characteristic impedance ZM in Eqs. 7 and 8 for a uniform material is a property characterizing the material and takes complex values which do not depend on the measuring circuit. The characteristic impedance is related to the maximum amplitudes of voltage V0 and current I0 for incident and reflected waves along the propagation direction x. The characteristic impedances of transmission lines for measuring circuits take real values within the design of the equipment even though they could take complex values.

The input impedance Zl(x) of the transmission lines at a certain position from port 1 in a circuit shown by Fig. 1 is defined by Eq. 9 and is apparently dependent upon the measuring circuit. The input impedance at x is also related to the maximum values, with respect to time, of voltage and current of either the incident or the reflected wave.20

$Zl(x)=V(x)I(x)=ZlejωLlClx+RMe−jωLlClxejωLlClx−RMe−jωLlClx$
(9)

Ll and Cl are the distributive inductance and capacitance for transmission lines, respectively, and they are usually real numbers. ω is the circular frequency of the incident microwave; RM in Eq. 9 is a constant which is independent of the position in the material. In material science it is defined as the reflection loss of the measured material at the interface between the transmission lines and the material, while in microwave electronics it is known as return loss which is represented by Γ.20 Using Eqs. 5 and 7, Eq. 9 can be rewritten as Eq. 10 for x < 0 in Fig. 1.

$Zl(x)=μlεlej2πvcεrμrx+RMe−j2πvcεrμrxej2πvcεrμrx−RMe−j2πvcεrμrx$
(10)

εr and μr in Eq. 10 are the relative permittivity and permeability for transmission lines, respectively, and they are usually real numbers. ν is the frequency of the incident microwave. The input impedance of the lossless transmission lines at x = 0 can be obtained from Eq. 10.

$Zl(0)=ZM=Zl1+RM1−RM$
(11)

The reflection loss RM for the material can be obtained from Eq. 11.

$RM=Zl(0)−ZlZl(0)+Zl=Zl(0)Zl−1Zl(0)Zl+1=ZMZl−1ZMZl+1$
(12)

From Eq. 12 if ZM = Zl, then RM = 0 which means there is no reflection within a uniformed material and that reflection only takes place at an interface. As ZM increases from 0, reflection loss will increase when ZM < Zl, and attain a maximum value at ZM = Zl, then decrease as ZM > Zl. Reflection loss for a material given by Eq. 12 is dependent on the apparatus used since the calculation involves Zl. But it is suitable to characterize materials within a series of systematic measurements. When the material studied is referenced to free-space at x = 0, instead of transmission lines in Fig. 1, the relative values of permittivity εr(free-space) and permeability μr(free-space) are both approximately equal to 1. Thus, only the relative permittivity and permeability of the absorbing material are required in Eq. 13 since $μ0ε0$ can been canceled from ZM and Zl. Using dB units and from Eqs. 5 and 7, Eq. 12 can be rewritten as Eq. 13. The reflection loss calculated from Eq. 13 is related to the free-space instead of to the apparatus used, thus Eq. 13 is more useful in practice.

$RM=20logμ0μrε0εr−μ0ε0μ0μrε0εr+μ0ε0=20logμrεr−1μrεr+1$
(13)

From Eqs. 10 and 12 we obtain

$Zl(x)=μlεlej2πvcεrμrx+RMe−j2πvcεrμrxej2πvcεrμrx−RMe−j2πvcεrμrx=μlεl(ZM+Zl)ej2πvcεrμrx+(ZM−Zl)e−j2πvcεrμrx(ZM+Zl)ej2πvcεrμrx−(ZM−Zl)e−j2πvcεrμrx=ZlZMcosh(j2πvcεrμrx)+Zlsinh(j2πvcεrμrx)ZMsinh(j2πvcεrμrx)+Zlcosh(j2πvcεrμrx)=ZlZM+Zltanh(j2πvcεrμrx)Zl+ZMtanh(j2πvcεrμrx)$
(14)

It is apparent from Eq. 14 that Zl(0) (= ZM) should always be used to calculate Zl(x) within the transmission lines. Zl(0) equals the characteristic impedance ZM. Although the input impedance of the transmission line Zl(x) changes with x, the same characteristic impedance ZM is always used in Eq. 14.20 Equation 14 is an expression with reference to the connection at x = 0 in Fig. 1 for x < 0. If we re-express Eq. 14 as Eq. 15 with reference to the connection at x = d in Fig. 1 for 0 < x < d, then ZM and Zl switch places.

$ZM(x)=ZMZL+ZMtanh(j2πvcεrμrx)ZM+ZLtanh(j2πvcεrμrx)$
(15)

εr and μr in Eq. 15 are the relative permittivity and permeability for the material, respectively, and they are usually complex numbers. A metal-backed ferrite structure can be viewed as short-circuiting the transmission line with a load ZL = 0 at x = d in Fig. 1 and the input impedance at x = d can be defined in eq. 16.21

$ZM(d)=ZMtanh(j2πvcdεrμr)=μ0μrε0εrtanh(j2πvcdεrμr)=Z0μrεrtanh(j2πvcdεrμr)$
(16)

We have found in the literature many examples where the method used to calculate reflection loss is not rigorous. For example, the one-port formulae Eq. 12 has often been used in two-port measurement. This is not a serious problem since the measured material parameter should be essentially the same even though the derivations involved are different.

Another problem is that the impedance at x = d calculated from Eq. 16 for ZM(d) has been used inappropriately in several publications as the characteristic impedance ZM in Eq. 12 to calculate reflection loss RM.6–14 It should be noted that RM is characteristic of the material studied and should be independent of x. However, in the commonly used incorrect method RM is often reported5,15,16 as a variable with respect to x, which results in an artificial result. Indeed, from Eq. 15 we can obtain Eq. 17 and then Eq. 18.

$ZM(x)=μMεMej2πvcεrμrx+RMe−j2πvcεrμrxej2πvcεrμrx−RMe−j2πvcεrμrx=ZM1+RMe−2j2πvcεrμrx1−RMe−2j2πvcεrμrx$
(17)
$RMe−2j2πvcεrμrx=ZM(x)−ZMZM+ZM(x)$
(18)

When the reference point is at x = d where the impedance changes abruptly, then x in Eq. 18 should be replaced by dx and ZM(d) becomes ZL. Thus, Eq. 18 is consistent with Eq. 12. Note that the input impedance ZM(x) is different from the characteristic impedance ZM. ZM(x) is a function of x where ZM is a constant complex number in Eq. 17 especially for lossless material. Lengthening lossless material by a section dx as shown in box A from Fig. 1 does not affect the value of the characteristic impedance, as calculated from Eq. 19 where the impedance 1/jωCMdx is first shunted with ZM and then connected with impedance jωLMdx in series20 or as calculated from Eq. 20 where the impedance (ZM + jωLMdx) is shunted with impedance 1/jωCMdx.22

$ZM+dZ=jωLMdx+ZM1jωCMdxZM+1jωCMdx=jωLMdx+ZM(ZMjωCMdx+1)−1=jωLMdx+ZM−ZM2jωCMdx=ZM$
(19)
$ZM+dZ=(ZM+jωLMdx)1jωCMdx(ZM+jωLMdx)+1jωCMdx=(ZM+jωLMdx)[1−jωCMdx(ZM+jωLMdx)]=ZM$
(20)

Equation 19 or 20 is valid for lossless material where CM and LM are real numbers. However, for microwave absorbing material ZM in Eq. 17 is still a constant even though the permittivity and permeability of the material are complex numbers, thus the reflection loss calculated from Eq. 12 at x = 0 is the same as that from Eq. 18 at x = d since the characteristic impedance ZM is a physical quantity characterizing material.

It has been shown Ref. 6 that the plot drawn from the reflection loss calculated at x = d with the conventional method is not the same as that from s11, which is an indication that there are problems with the calculations since there should be no difference between reflection loss and scattering parameter s11.17,18 The relationship between RM and s11 is proved below by Eq. 33.

When the load ZL in Fig. 2a is equal to Z1, the power output at port 1 is maximum and all the energy from port 1 is consumed by the load. IL must be the same as I1 in the series circuit shown by Fig. 2a. However, if ZL is not equal to Z1, then IL = V1/ZL is not equal to I1 = V1/Z1. To fix this discrepancy, a reflected wave is needed which validates Eqs. 21 and 22.

$V+I+=−V−I−=Z1$
(21)
$V++V−I++I−=V1I1=V1IL=ZL$
(22)
FIG. 2.

(a) one-port circuit. (b) two-port circuit. Port 1 is enclosed in boxes indicated with bold lines. R is the reflection loss. a and b are the wave amplitudes related to the calculation of scattering parameters. The figure in box A is equivalent to those presented in boxes B and C. asource and bsource can be normalized to a1 and b1. Zi, Vi, and Ii are the characteristic impedance, amplitudes of voltage and current for port i, (i = 1, 2).

FIG. 2.

(a) one-port circuit. (b) two-port circuit. Port 1 is enclosed in boxes indicated with bold lines. R is the reflection loss. a and b are the wave amplitudes related to the calculation of scattering parameters. The figure in box A is equivalent to those presented in boxes B and C. asource and bsource can be normalized to a1 and b1. Zi, Vi, and Ii are the characteristic impedance, amplitudes of voltage and current for port i, (i = 1, 2).

Close modal

V+ and I+ are the maximum amplitudes of voltage and current along the direction of propagation respectively for the forward wave, and V- and I- are the maximum amplitudes for the reflected wave with negative values.20 The reflection coefficient RM in Eq. 9 is defined as the ratio of the maximum amplitudes of the reflected and incident wave.

$R=V−V+=I−(−Z1)I+Z1=−I−I+$
(23)

RM is the reflection loss characteristic of a material and need not be reported with reference to the thickness of the material, despite the fact that in the literature the value of RM is often artificially adjusted by the thickness of the material.5,15 Combining Eqs. 22 and 23, we obtain Eq. 24 which is consistent with Eq. 11.

$ZL=V++V−I++I−=V+(1+R)I+(1−R)=Z1V+(1+R)Z1I+(1−R)=Z11+R1−R$
(24)

The outgoing voltage from the source is related to the output voltage V1 by Eq. 25 and the return voltage to the source is related to the output voltage V1 by Eq. 26. The incident and the reflected amplitudes asource and bsource in boxes B and C from port 1 in Fig. 2b have been normalized in Eqs. 25 and 26.

$a1=asource=asource′2Z1+Z1*=I1Z1+V12Z1+Z1*$
(25)
$b1=bsource=bsource′2Z1+Z1*=V1−I1Z1*2Z1+Z1*$
(26)

or

$a1+b1*=2V12(Z1+Z1*)=2(V++V−)Z1+Z1*$
(27)
$a1−b1*=I1(Z1+Z1)2(Z1+Z1*)=(I++I−)(2Z1)2(Z1+Z1*)$
(28)

The impedance for the port is usually a real number. Thus

$a1+b1=(V++V−)Z1$
(29)
$a1−b1=(I++I−)Z1=(V+Z1−V−Z1)Z1=V+Z1−V−Z1$
(30)
$a1=V+Z1$
(31)
$b1=V−Z1$
(32)

The definition of a1 in Eqs. 25 and 31 represents the incident wave a1 and the definition of b1 in Eqs. 26 and 32 represents the reflected wave.

Smith charts relate reflection loss to the impedance of load. In a Smith chart s11 is the reflection loss and the impedance of load is calculated from s11. The definition of s11 in Eq. 33 is equivalent to the reflection loss RM from Eq. 9 or 12 since Z1 is a real number and thus RM can be reported directly from the scattering s11.17,18 The reflection loss s11 is the ratio of b1 and a1 measured at port 1 with port 2 properly terminated in its normalizing impedance where a2 = 0.

$s11=b1a1a2=0=V1−I1Z1*V1+I1Z1V2=−I2Z2=I1ZL−I1Z1*I1ZL+I1Z1V2=−I2Z2=ZL−Z1*ZL+Z1V2=−I2Z2=RM$
(33)

Equation 33 is consistent with Eq. 12 or 18 and the value obtained from the equation is referenced to the measuring circuit. In a short-circuit transmission line ZL = 0 and reflection loss |R| = |s11| = |-1| = 1. As a consequence there is total reflection. In the open-circuit transmission line, ZL = ∞ and R = s11 = 1. Again, there is total reflection.

Power is dependent on the characteristic impedance. The normalized a1 and b1 are related to incident and reflection powers (Eqs. 34 and 35), respectively. The maximum power Pmax that can be delivered by the source occurs when the load ZL is a conjugate match of the source $Z1*$ (Fig 2a) where no reflection occurs.23 The actual power output from port 1 is related to

$a1a1*=a12=(I1Z1+V1)(I1Z1+V1)*2(Z1+Z1*)=(I1Z1+I1ZL)(I1Z1+I1ZL)*2(Z1+Z1*)$
(34)

For sinusoidal signals, the mean square power r. m. s. value of the phasor and their peak values are related by $2$. The power dissipated by the load ZL is related to

$a1(a1)*−b1(b1)*=a12−b12=I1(Z1+ZL)I1*(Z1+ZL)*2(Z1+Z1*)−I1(ZL−Z1*)[ZL*−Z1]I1*2(Z1+Z1*)=I1(Z1Z1*+Z1ZL*+ZLZ1*+ZLZL*)I1*2(Z1+Z1*)− I1(ZLZL*−ZLZ1−Z1*ZL*+Z1*Z1)I1*2(Z1+Z1*)=I1[Z1(ZL+ZL*)+Z1*(ZL+ZL*)]I1*2(Z1+Z1*)=I1(ZL+ZL*)I1*2$
(35)

The product $I(I*)$ in Eq. 35 is a real number since the imaginary parts have been canceled. Voltage and current cannot be measured from microwave frequency since the effects of distributed inductance and capacitance cannot be neglected. In microwave electronics, a standing wave with maximum amplitude V+ + V- and minimum amplitude V+ - V- is generated from the incident and reflected waves along the transmission lines. Thus it is impractical to measure voltage in microwave networks. However, power is easier to measure. |a|2/2 and |b|2/2 represent the power in the forward and backward propagating waves respectively. The definitions of a and b allow propagation to be defined in terms of maximum wave amplitudes that are directly related to the power in the wave. In microwave electronics, analyses can be carried out using scattering parameters and reflection loss because they are defined in a and b.

If a and b are defined in a way that the imaginary part b1a1$*$ - (b1a1$*$)$*$ is not zero then the r. m. s. power is given by Eq. 38.24,25 If we define Eq. 36 from Eq. 28

$I=I++I−=2(a1−b1)Z1+Z1*$
(36)

and from Eq. 8

$V=Z1I++I−(−Z1)=Z12(a1+b1)Z1+Z1*$
(37)

then

$P=Re12π∫0t=1/vVI*sin2(2πvt−2πvcx)d(2πvt)=12Re(VI*)=Z1(a1+b1)Z1+Z1*[a1−b1Z1+Z1*]*=Z1[(a1a1*−b1b1*)+(b1a1*−a1b1*)]Z1+Z1*=Re(Z1)(a1a1*−b1b1*)2Re(Z1)+Im(Z1){[b1a1*−(b1a1*)*]}2Re(Z1)=12(a12−b12)+Im(b1a1*)Im(Z1)Re(Z1)$
(38)

Re and Im in Eq. 38 take the real and imaginary parts of a complex number, respectively.

The permeability of the material cannot be obtained from single port measurement since only the capacitance is effective in Box A of Fig. 1 for the interaction of microwaves and the material. While for a two-port measurement microwaves are transmitted through the material and thus both inductance and capacitance should be considered. Although permeability cannot be obtained from single port measurement, essentially the same value for a common measureable parameter should be obtained from both single and two-port measurements. However their calculations are different. Rin can be obtained by Eqs. 3944 from a two port measurement system represented in Fig. 2b.

$Rload=aloadbload=a2b2$
(39)

From Eq. 39

$b1=s11a1+s12a2=s11a1+s12Rloadb2$
(40)
$b2=s21a1+s22a2=s21a1+s22Rloadb2$
(41)

Rearranging Eqs. 40 and 41

$b1a1=s11+s12Rloadb2a1$
(42)
$b2a1=s21+s22Rloadb2a1$
(43)

Inserting Eq. 43 into Eq. 42

$Rin=b1a1=s11+s12Rloads211−s22Rload=s11−(s11s22−s12s21)Rload1−s22Rload$
(44)

Similarly, we obtain Rout from Eqs. 4549.

$Rsource=asourcebsource=a1b1$
(45)

From Eq. 45

$b1=s11a1+s12a2=s11Rsourceb1+s12a2$
(46)
$b2=s21a1+s22a2=s21Rsourceb1+s22a2$
(47)

Rearranging Eqs. 46 and 47

$b1a2=s11Rsourceb1a2+s12$
(48)
$Rout=b2a2=s21Rsourceb1a2+s22=s21Rsources121−s11Rsource+s221−s11Rsource1−s11Rsource=s22−(s22s11−s21s12)Rsource1−s11Rsource$
(49)

Reflection loss is apparatus dependent. Equations 44 and 49 should be used in two-port measurement rather than Eqs. 12 and 16. When Rload = 0 we obtain Rin = s11 from Eq. 44 and when Rsource = 0, we obtain Rout = s22 from Eq. 49, a process which is consistent with the definition of reflection loss given by Eq. 33. Rin might not be equal to Rout since the sample holder (such as Keysight 85055A Type-N 50 Ω Verification Kit) is not usually symmetrically designed.

With the theoretical background presented in previous sections, the problems with the conventional calculations of reflection loss can be addressed. As shown by Fig. 3 microwave waves will be absorbed when they travel into wave absorbing material. The deeper a microwave travels, the more waves should be absorbed. It is misleading to state that reflection loss has a maximum5,15,16 with respect to sample thickness. It is apparent that the same program in calculating reflection loss is generally used since the same type of misleading results has appeared among publications from different research groups. These errors might have been avoided if the calculations were implemented in Excel especially with the build-in functions for complex numbers. Furthermore, the reflection loss at x = d should be measured from port 2 in a two-port measurement as shown by Fig. 2. Here, the interface at x = d becomes the reference plane and x in Eqs. 15, 17, and 18 should be replaced by (dx).

FIG. 3.

A slab of microwave absorbing material with interfaces at x = 0 and d. Coordinates before and after an interface or a plane are indicated with – and + respectively. ai and bi represent incident and reflected beams, respectively.

FIG. 3.

A slab of microwave absorbing material with interfaces at x = 0 and d. Coordinates before and after an interface or a plane are indicated with – and + respectively. ai and bi represent incident and reflected beams, respectively.

Close modal

Since the permittivity and permeability of microwave absorbing material are complex numbers, $tanh(j2πvcdεrμr)$ = tanh(jy) in Eq. 16 takes values from 0 to 1 as d increases from 0. When the input impedance ZM(d) in Eq. 16 takes the place of the characteristic impedance ZM in Eq. 12 and if ZM is larger than Zl when referenced to the apparatus or larger than 1 when referenced to open-space, then there will be total absorption when ZM(d) = Zl or 1 as tanh(y) increases from 0 to 1 (Fig. 4). However, this is not a proof that RM can attain a maximum value with respect to the sample thickness but rather evidence that using ZM(d) in place of ZM in Eq. 12 is inappropriate.

FIG. 4.

This plot reveals the artificial aspect of the conventional calculation for reflection loss R. The plot is drawn from Eq. 12 in decibel units. ZM = 1.1 Ω and Zl = 1 Ω. In literature, ZM(x) calculated from Eq. 16 has been used as ZM in Eq. 12. The Mathematica file has been provided as a supplementary material.

FIG. 4.

This plot reveals the artificial aspect of the conventional calculation for reflection loss R. The plot is drawn from Eq. 12 in decibel units. ZM = 1.1 Ω and Zl = 1 Ω. In literature, ZM(x) calculated from Eq. 16 has been used as ZM in Eq. 12. The Mathematica file has been provided as a supplementary material.

Close modal

According to Eq. 12, there are reflections at interfaces at x = 0 and d (Fig. 3) since there is an abrupt change in characteristic impedance at the interface. There is no interface at x = x1 since there is not an abrupt change of characteristic impedance on both sides of the plane at x1 even though ZM(x1) is not the same as ZM(x2). Thus, microwaves penetrated into a material will not be reflected along x = 0 to x = d since the wave travels in the same medium, which means that ZM in Eq. 12 should not be replaced by ZM(x) to calculate the reflection loss at x between 0 and d. Even though the ratios of bM and aM are not the same at x1 and x2, the characteristic impedances at x- and x+ are the same since microwaves are absorbed but not reflected within the same material.

$|RMe−2j2πvcεrμrx|$ is a constant for lossless material thus Eq. 18 shows that there is no reflection within lossless material. The result is consistent with that obtained from Eq. 12. $j2πvcεrμrx$ is a complex number for microwave absorbing material since the relative permittivity and permeability are both complex numbers. However, within the material there is no interface, and therefore $|RMe−2j2πvcεrμrx|$ should not be a property characterizing the reflection ability of microwave absorbing material since there is no reflection in the material. It can only be a property for the circuit which characterizes the amount of the incident radiation “absorbed” into the material. More precisely, it does not characterize absorption but the amount of microwave transmitted from the interface into the material. Thus, reflection loss should only be reported from Eq. 12 or 33 at x = 0 or Eq. 18 at d, and not with reference to Eq. 16. What is more, in contrast with the lossless material, RM calculated from its definition (Eq. 23) is not a constant within a microwave absorbing material. If the reflected beam at x1 was reflected from x1, then RM could be a constant. However, the reflected beam at x1 is not the wave reflected at x1 but the wave from x = d. And the wave is absorbed as it travels from x1 to d and is then reflected from d to x1. Thus, the values of RM calculated from Eq. 23 at x1 and x2 are not the same within a microwave absorbing material. However, microwaves with various intensities can indeed be reflected at x = 0 and d, thus the reflection loss measured at x = 0- or d- from port 1 should be the same as that measured at d+ from port 2 and thus both Eqs. 12 and 18 are valid to apply in the calculations. The absorptions for incident and reflected waves are of the same proportion at x = d which means that the absorption has not changed the characteristic impedance of the material at d.

It is common practice in microwave absorption studies to (inappropriately) use ZM(x) calculated from Eq. 16 at x = d as the characteristic impedance ZM in Eq. 12 to calculate reflection loss RM.26 In fact it would be more correct to use Eq. 18 to calculate the reflection loss within a material even though this process would still be problematic because the reflection loss at x = d is not a measured value but a calculated value in the conventional method, i.e., when the permittivity and permeability of a material have been measured, the reflection loss RM at x = d was then calculated from Eqs. 12 and 16. Thus, the sample thickness d used in the calculation functions as an artificial means to magnify the absolute value of absorption.

Ferrite and its composites are composed of individual grains. Reflections occur at interface between grains or particles. Should this justify the conventional method for calculating reflection loss? Actually, it has been proved that transmission line theory applies to a wide variety of materials by considering the average properties, such as the accumulative εr from the reflections of different grains,18 in a uniform material and thus the materials can be considered as a single phased material with uniform characteristic impedance in transmission line theory, whether the material consists of ferrite grains or its composites. The results obtained using the macro properties of a uniform material should not be replaced by those from the next level of micro theory when grain boundaries are taken into consideration.

It can be seen from Eq. 12 that the reflection loss cannot be greater than 1. However, an algorithm mistake is imbedded in a program available in the marketplace since we find it sometimes gives values of reflection loss greater than 1. If the algorithm shown in Eq. 50 is adopted, then the artificial result that RM greater than 1 might be obtained when Eq. 51 or 52 is satisfied.

$RM=Zr+jZj−1Zr+jZj−1=(Zr−1+jZj)(Zr+1−jZj)(Zr+1)2+Zj2=Zj2−1+Zj2+j[(Zr+1)Zj−(Zr−1)Zj](Zr+1)2+Zj2=Zr2−1+Zj2+j2ZrZj(Zr+1)2+Zj2$
(50)
$[Zr2−1+Zj2]2+(2ZrZj)2=Zr4+1−2Zr2+Zj4+2Zj2(Zr2−1)+4(ZrZj)2>[(Zr+1)2+Zj2]2=(Zr2+1+2Zr)2+Zj4+2Zj2(Zr+1)2=Zr4+1+6Zr2+4Zr3+4Zr+Zj4+2Zj2Zr2+4Zj2Zr+2Zj2$
(51)
$4Zj2(Zr2−Zr)>8Zr2+4Zr3+4Zr$
(52)

However, with the algorithm shown by Eq. 53, RM can never greater be than 1.

$RM=ZMZl−1ZMZl+1=ZMZl−1ZMZl+1=Zr+jZj−1Zr+jZj+1≤1$
(53)

Although the choice of algorithm might affect the result the calculated values from the two algorithms will not different very much since the imaginary part of characteristic impedance is usually not large and its real part is positive. However, there might be a more significant problem, as indicated in Fig. 5a, in which the largest peak might be missed because the measuring step for microwave frequency is too large. The largest peak missed in Fig. 5a does appear in Fig. 5b when the interpolation technique is used. The maximum peak occurs when the characteristic impedance ZM jumps from 0.5 to 1.8.

FIG. 5.

(a) Reflection loss versus microwave frequency for NSm0.12Fe1.88O4 synthesized in our laboratory. (b) Characteristic impedance ZM versus microwave frequency. The peak for reflection loss between 1.188 and 1.205 Hz missed in (a) has been restored in the insert in (b) by using the interpolation technique. The Excel file for the data in the figure has been provided as a supplementary material.

FIG. 5.

(a) Reflection loss versus microwave frequency for NSm0.12Fe1.88O4 synthesized in our laboratory. (b) Characteristic impedance ZM versus microwave frequency. The peak for reflection loss between 1.188 and 1.205 Hz missed in (a) has been restored in the insert in (b) by using the interpolation technique. The Excel file for the data in the figure has been provided as a supplementary material.

Close modal

A property characterizing a material should be intrinsic to the material itself and be independent of the apparatus and sample thickness. Reflection loss or the scattering parameter s11 can be used to characterize a material even though its values are different with reference to the apparatus and free-space. Reflection loss calculated with reference to open-space does not rely on the apparatus and is therefore useful in practice. However, it is convenient to report reflection loss from s11 though this parameter references to the measuring circuit.

It is interesting to note that the problems with the conventional method for calculating reflection loss have not been addressed over many years, and the current incorrect method has had considerable influence, indeed it has almost become the standard method. The problems are caused by ignoring the fact that reflection only takes place at the boundary of different media with different characteristic impedances. It is suggested that the problems indicate a great gap between the expertise of material scientists and microwave engineers. Thus, the transmission line theory has been briefly covered here in an easily understandable way.

A general lesson from this work is that in research, returning to the roots and fundamentals of an established theory is at times as important as pushing the boundaries with state-of-the-art equipment on hot trendy topics. Fundamental principles are still tremendously valuable and relevant for research in modern days and as we have shown here, improvements can often be made to theories previously held to be fit for purpose. If new light is shone on traditional theories as in this work, the results are sometimes passed off as too basic to be considered as important. In fact, sustainability in the history of science and informative should be the criterions for the significance of research.27

See supplementary material for Figs. 4 and 5 have been provided.

In an infrared spectrum the induced dipolar moments for chemical bonds are important since absorptions occur between vibrational energy levels. Microwave absorption occurs because of interactions between the electromagnetic field and the rigid rotators of permanent or induced dipolar moments whereas the induced dipolar moments mainly come from procession other than excitations from vibration states. The absorption peak with the maximum intensity from the rotations and oscillations of rigid rotors is characterized by resonance absorption. Although all absorptions originate from resonance, the weaker absorptions have different characteristics and have been defined as forced non-resonance absorptions.18,19 This is because they occur only from a few states and the energy gaps between rotation levels are small. Thus, they are also related with state distributions involving average behavior leading to broad and weak absorption bandwidth. With a classical view, energy is consumed from resistance to rotation or oscillation in these forced non-resonance absorptions. However, a more detailed theoretical study for microwave absorption should use Maxwell’s equations19 and quantum theory.20 The mathematical techniques necessary for more informative theoretical research are included in the following appendices. Subsection 1 of the  Appendix gives proofs of the vector identities involving the Nabla operator. Three methods to obtain gradient, divergence, Laplacian, and curl operators in general orthogonal coordinate systems are compared in Subsection 2 of the  Appendix. These include a simple method involving few mathematical skills, an intermediate method using identities introduced in Subsection 1 of the  Appendix, and an advanced method using the rarely used concept of reciprocal vectors. In Subsection 2g of the  Appendix, Gauss’s and Stokes’s theorems are related to Green’s theorem in a novel way via the expressions of gradient, divergence, Laplacian, and curl operators. The discussion also reveals the physics behind the mathematics used in the three methods introduced in this section. Subsections  2–4 of the  Appendix are indispensable in electromagnetism and quantum theory. Although the contents in these Appendices are traditional28–33 the presentations are novel.

1. Vector identities involve the Nabla operator

This Appendix describes the mathematical skills34 needed in the development of Maxwell’s equations. i, j, k are unit vectors in the Cartesian coordinate system. The Nabla or del operator is also known as the Hamilton or ∇ operator.

$∇=i∂∂x+j∂∂y+k∂∂z$
(A1)

Some of the useful identities include

$∇×u(x,y,z)c=∇u(x,y,z)×c$
(A2)
$∇×[∇u(x,y,z)]=0$
(A3)
$∇⋅[∇×A(x,y,z)]=0$
(A4)
$∇⋅[u(x,y,z)A(x,y,z)]=∇u(x,y,z)⋅A(x,y,z)+u(x,y,z)∇⋅A(x,y,z)$
(A5)
$∇×[u(x,y,z)A(x,y,z)]=∇u(x,y,z)×A(x,y,z)+u(x,y,z)∇×A(x,y,z)$
(A6)
$∇⋅(A×B)=B⋅(∇×A)−A⋅(∇×B)$
(A7)
$∇×(∇×A)=∇(∇⋅A)−∇2A$
(A8)
$∇×(A×B)=(B⋅∇)A−(A⋅∇)B−B(∇⋅A)+A(∇⋅B)$
(A9)
$∇[A(x,y,z)⋅B(x,y,z)]=A×(∇×B)+(A⋅∇)B+B×(∇×A)+(B⋅∇)A$
(A10)

Bold letters indicate vector functions. c is a constant vector. u and f are scalar functions. The proofs of the above identities are usually omitted for simplicity so they are presented here. Equations A2, A3, and A4 can be proved by Equations A11, A12, and A13, respectively.

$∇×uc=ijk∂∂x∂∂y∂∂zucxucyucz=ijk∂u∂x∂u∂y∂u∂zcxcycz=∇u×c=i(∂u∂ycz−∂u∂zcy)−i(∂u∂xcz−∂u∂zcx)+k(∂u∂xcy−∂u∂ycx)$
(A11)
$∇×(∇u)=ijk∂∂x∂∂y∂∂z∂u∂x∂u∂y∂u∂z=0$
(A12)
$∇⋅(∇×A)=∂∂x∂∂y∂∂z∂∂x∂∂y∂∂zAxAyAz=∂∂x(∂Az∂y−∂Ay∂z)−∂∂y(∂Az∂x−∂Ax∂z)+∂∂z(∂Ay∂x−∂Ax∂y)=0$
(A13)

Proof of Equation A5

In this proof, subscript c indicates that the function is kept constant.

$∇⋅(uA)=∇⋅(ucA)+∇⋅(uAc)=uc∇⋅A+(∇⋅u)Ac=u∇⋅A+(∇u)⋅A$
(A14)

Equation A6 can be proved similarly with Eq. A2.

$∇×(uA)=∇×(uAc)+∇×(ucA)=ijk∂u∂x∂u∂y∂u∂zAxAyAz=ijku∂∂xu∂∂yu∂∂zAxAyAz=∇u×A+u∇×A$

Proof of Equation A7

$∇⋅(A×B)=∇⋅(A×Bc)+∇⋅(Ac×B)=∇⋅(A×Bc)−∇⋅(B×Ac)=(∇×A)⋅Bc−(∇×B)⋅Ac=(∇×A)⋅B−(∇×B)⋅A$
(A15)

Proof of Equation A8

$∇×(∇×A)=ijk∂∂x∂∂y∂∂z∂∂y∂∂zAyAz∂∂z∂∂xAzAx∂∂x∂∂yAxAy=[∂∂y(∂Ay∂x−∂Ax∂y)−∂∂z(∂Ax∂z−∂Az∂x)]i+[∂∂z(∂Az∂y−∂Ay∂z)−∂∂x(∂Ay∂x−∂Ax∂y)]j+ [∂∂x(∂Ax∂z−∂Az∂x)−∂∂y(∂Az∂y−∂Ay∂z)]k=∂2(Axj+Ayi)∂x∂y+∂2(Ayk+Azj)∂y∂z+∂2(Axk+Azi)∂x∂z− ∂2(Ayj+Azk)∂x2−∂2(Axi+Azk)∂y2−∂2(Axi+Ayj)∂z2=(i∂∂x+j∂∂y+k∂∂z)(∂Ax∂x+∂Ay∂y+∂Az∂z)−(∂2∂x2+∂2∂y2+∂2∂z2)(Axi+Ayj+Azk)=∇(∇⋅A)−∇2A$
(A16)

or

$C×(B×A)=B(C⋅A)−(C⋅B)A$
$∇×(∇×A)=∇(∇⋅A)−(∇⋅∇)A=∇(∇⋅A)−∇2A$
(A17)

Proof of Equation A9

$∇×(A×B)=∇×(Ac×B)+∇×(A×Bc)=[Ac(∇⋅B)−(Ac⋅∇)B]+[(Bc⋅∇)A−(∇⋅A)Bc]=A(∇⋅B)−(A⋅∇)B+(B⋅∇)A−B(∇⋅A)$
(A18)

Proof of Equation A10

$∇(A⋅B)=∇(Ac⋅B)+∇(A⋅Bc)=∇(Ac⋅B)+∇(Bc⋅A)=[Ac×(∇×B)+(Ac⋅∇)B]+[Bc×(∇×A)+(Bc⋅∇)A]=A×(∇×B)+(A⋅∇)B+B×(∇×A)+(B⋅∇)A$
(A19)

Some examples related to the identities introduced above

$r=xi+yj+zk$
(A20)
$∇r=(i∂∂x+j∂∂y+k∂∂z)x2+y2+z2=2xi+2yj+2zk2x2+y2+z2=rr$
(A21)
$∇⋅r=(i∂∂x+j∂∂y+k∂∂z)(xi+yj+zk)=3$
(A22)
$∇×r=ijk∂∂x∂∂y∂∂zxyz=0$
(A23)
$∇f(u,υ)=∂f∂u|υ∇u+∂f∂υ|u∇υ$
(A24)

Application of Eqs. A21 and 24

$∇f(r)=∂f∂r∇r=∂f∂rrr$
(A25)

Application of Eqs. A6, A23, and A25

$∇×[rf(r)]=f(r)∇×r+∇f(r)×r=∂f∂rrr×r=0$
(A26)

Application of Eqs. A6, A21, A23, and A25

$∇×(r−3r)=−3r−4rr×r+r−3∇×r=0$
(A27)

2. Comparison of calculations for gradient, divergence, Laplacian, and curl in non-Cartesian systems

The conversions of gradient, divergence, Laplacian, and curl operators between orthogonal coordinate systems are usually discussed intuitively.28 The issues are presented here in general terms. The three methods introduced are essentially the same except for the methods of collecting terms. Method 2 in Subsections 2b and 2c of the  Appendix is the simplest since it only involves elementary mathematical skills. Method 1 in Subsection 2a of the  Appendix is intermediate since vector identities involving the Nabla operator in Subsection 1 of the  Appendix have been used. Method 3 in Subsection 2e of the  Appendix is more advanced and the concept of reciprocal vectors used is novel. Applications are introduced in Subsection 2g of the  Appendix to reveal the physical meaning behind the mathematics involved in the methods. This Appendix shows that every method has its own merits and should not be judged by whether it is intuitive or abstract, simple or involved.

Vectors e1, e2, and e3 are mutually perpendicular unit vectors in three-dimensional space and q1, q2, and q3 are the corresponding three independent coordinates.

$e1=e2×e3;e2=e3×e1;e3=e1×e2$
(A28)

s is the length of an arc in the general coordinate system. Hi is the Lame coefficient. With reference to the Cartesian coordinate system

$ds=(dx)2+(dy)2+(dz)2$
(A29)
$dsi=[∂x(q1,q2,q3)∂qidqi]2+[∂y(q1,q2,q3)∂qidqi]2+[∂z(q1,q2,q3)∂qidqi]2=(∂x∂qi)2+(∂y∂qi)2+(∂z∂qi)2dqi=Hidqi; i=1,2,3$
(A30)

In Equation A30, all the qj(x, y, z) values are constant when j is not equal to i, i.e. dqj = 0 for j ≠ i. The gradient to qi(x, y, z) = constant is given by Eq. A31 which will be proved by Eq. A71 or A101.

$∇qi=∂qi∂siei=∂qi∂eiei=1Hiei; i=1,2,3$
(A31)
a. Method 1 for gradient, divergence, Laplacian, and curl in a general and orthogonal coordinate system

From the definition of gradient, we obtain A33 from Eqs. A24 and A31.

$∇f(q1,q2,q3)=∂f∂q1∇q1+∂f∂q2∇q2+∂f∂q3∇q3=1H1∂f∂q1e1+1H2∂f∂q2e2+1H3∂f∂q3e3$
(A32)

The gradient of f is defined as the directional derivation of f along an arc s. From Eq. 30 we have dsi = Hidqi.

$∇f(q1,q2,q3)=dfds=∂f∂q1∇q1+∂f∂q2∇q2+∂f∂q3∇q3=e1∂f∂q1q2,q3∂q1∂s1q2,q3+e2∂f∂q2q1,q3∂q2∂s2q1,q3+e3∂f∂q3q1,q2∂q3∂s3q1,q2=e1∂f∂q1∂q1∂s1+e2∂f∂q2∂q2∂s2+e3∂f∂q3∂q3∂s3=e1∂f∂s1+e2∂f∂s2+e3∂f∂s3=e1H1∂f∂q1+e2H2∂f∂q2+e3H3∂f∂q3$
(A33)

From Eqs. A3, A5, A7, A28, and A31

$∇⋅A=∇⋅(A1e1+A2e2+A3e3)=∇⋅[A1(e2×e3)+A2(e3×e1)+A3(e1×e2)]=∇⋅[A1H2H3(∇q2×∇q3)+A2H3H1(∇q3×∇q1)+A3H1H2(∇q1×∇q2)]=A1H2H3∇⋅(∇q2×∇q3)+∇(A1H2H3)⋅(∇q2×∇q3)+ A2H3H1∇⋅(∇q3×∇q1)+∇(A2H3H1)⋅(∇q3×∇q1)+ A3H1H2∇⋅(∇q1×∇q2)+∇(A3H1H2)⋅(∇q1×∇q2)=A1H2H3(∇×∇q2)⋅∇q3−A1H2H3(∇×∇q3)⋅∇q2+∇(A1H2H3)⋅(∇q2×∇q3)+ 0+0+∇(A2H3H1)⋅(∇q3×∇q1)+0+0+∇(A3H1H2)⋅(∇q1×∇q2)={[∂∂q1(A1H2H3)]∇q1+[∂∂q2(A1H2H3)]∇q2+[∂∂q3(A1H2H3)]∇q3}⋅(∇q2×∇q3)+ {[∂∂q1(A2H3H1)]e1⋅(e3×e1)H1H3H1+[∂∂q2(A2H3H1)]e2⋅(e3×e1)H2H3H1+0}+ {0+0+[∂∂q3(A3H1H2)]e3⋅(e1×e2)H3H1H2}=1H1H2H3[∂∂q1(H2H3A1)+∂∂q2(H1H3A2)+∂∂q3(H1H2A3)]$
(A34)

From Eqs. A33 and A34

$∇2f=∇⋅[∇f]=∇⋅(∂f∂q1∇q1+∂f∂q2∇q2+∂f∂q3∇q3)=∇⋅(∂f∂q1e1H1+∂f∂q2e2H2+∂f∂q3e3H3)=1H1H2H3[∂∂q1H2H3H1∂f∂q1+∂∂q2H1H3H2∂f∂q2+∂∂q3H1H2H3∂f∂q3]$
(A35)

From Eqs. A6, A24, and A31

$∇×A=∇×(A1e1+A2e2+A3e3)=∇×(A1H1∇q1+A2H2∇q2+A3H3∇q3)=[A1H1(∇×∇q1)+∇(A1H1)×∇q1]+0+∂∂q1(A2H2)∇q1+∂∂q2(A2H2)∇q2+ ∂∂q3(A2H2)∇q3×∇q2+{0+[∂∂q1(A3H3)e1×e3H1H3+∂∂q2(A3H3)e2×e3H2H3+0]}={0+[0−e3H1H2∂(A1H1)∂q2+e2H1H3∂(A1H1)∂q3]}+[e3H1H2∂(A2H2)∂q1+0−e1H2H3∂(A2H2)∂q3]+ [−e2H1H3∂(A3H3)∂q1+e1H2H3∂(A3H3)∂q2]=1H1H2H3H1e1H2e2H3e3∂∂q1∂∂q2∂∂q3H1A1H2A2H3A3$
(A36)
b. The derivatives of e1, e2, and e3 with respect to q1, q2, and q3

These formulae will be used in method 2 in Subsection 3 of the  Appendix.

$∂e1∂q1=−e2H2∂H1∂q2−e3H3∂H1∂q3$
(A37)
$∂e1∂q2=e2H1∂H2∂q1$
(A38)
$∂e1∂q3=e3H1∂H3∂q1$
(A39)
$∂e2∂q1=e1H2∂H1∂q2$
(A40)
$∂e2∂q2=−e3H3∂H2∂q3−e1H1∂H2∂q1$
(A41)
$∂e2∂q3=e3H2∂H3∂q2$
(A42)
$∂e3∂q1=e1H3∂H1∂q3$
(A43)
$∂e3∂q2=e2H3∂H2∂q3$
(A44)
$∂e3∂q3=−e1H1∂H3∂q1−e2H2∂H3∂q2$
(A45)

Equations A46A50 are needed to prove the above equations via method 2.

$r=xi+yj+zk$
(A46)
$dr=ds$
(A47)

From Eqs. A30, A31, A46, and A47 we obtain Eq. A48 which is consistent with Eqs. A97, A107 and A108.

$∂r∂qi=∂x∂qii+∂y∂qij+∂z∂qik=∂si∂qiei=Hiei i=1,2,3$
(A48)

Multiplying both sides of Eq. A48 by i, j, and k, respectively

$ei⋅i=1Hi∂x∂qi;ei⋅j=1Hi∂y∂qi;ei⋅k=1Hi∂z∂qi$
(A49)

For orthogonal systems

$i=(e1⋅i)e1+(e2⋅i)e2+(e3⋅i)e3=e1H1∂x∂q1+e2H2∂x∂q2+e3H3∂x∂q3$
(A50)

There are similar equations for j and k.

Proof of Eq. A37 from Eqs. A48 and A50.

$∂e1∂q1=∂∂q1[1H1(∂x∂q1i+∂y∂q1j+∂z∂q1k)]=(∂x∂q1i+∂y∂q1j+∂z∂q1k)∂∂q1(1H1)+ 1H1(∂2x∂q12i+∂2y∂q12j+∂2z∂q12)+1H1(∂x∂q1∂∂q1i+∂y∂q1∂∂q1j+∂z∂q1∂∂q1k)=H1e1∂∂q1(1H1)+1H1∂2x∂q12(e1H1∂x∂q1+e2H2∂x∂q2+e3H3∂x∂q3)+ 1H1∂2y∂q12(e1H1∂y∂q1+e2H2∂y∂q2+e3H3∂y∂q3)+1H1∂2z∂q12(e1H1∂z∂q1+e2H2∂z∂q2+e3H3∂z∂q3)+0$
(A51)

From Equation A30

$∂x∂q1∂2x∂q12+∂y∂q1∂2y∂q12+∂z∂q1∂2z∂q12=H1∂H1∂q1=12∂H12∂q1$
(A52)

From Eqs. A30 and A48 we obtain

$∂2x∂q12∂x∂q2+∂2y∂q12∂y∂q2+∂2z∂q12∂z∂q2=∂∂q1(∂x∂q1∂x∂q2+∂y∂q1∂y∂q2+∂z∂q1∂z∂q2)− (∂x∂q1∂2x∂q1∂q2+∂y∂q1∂2y∂q1∂q2+∂z∂q1∂2z∂q1∂q2)=∂∂q1(H1e1⋅H2e2)−12∂H12∂q2=0−H1∂H1∂q2$
(A53)
$∂2x∂q12∂x∂q3+∂2y∂q12∂y∂q3+∂2z∂q12∂z∂q3=−H1∂H1∂q3$
(A54)

Inserting Eqs. A52A54 into Eq. A51

$∂e1∂q1=−e1H1∂H1∂q1+e1H12H1∂H1∂q1−e2H1H2H1∂H1∂q2−e3H1H3H1∂H1∂q3=−e2H2∂H1∂q2−e3H3∂H1∂q3$
(A55)

Proof of Eq. A38 from Eqs. A48 and A50

$∂e1∂q2=∂∂q2[1H1(∂x∂q1i+∂y∂q1j+∂z∂q1k)]=(∂x∂q1i+∂y∂q1j+∂z∂q1k)∂∂q2(1H1)+1H1(∂2x∂q1∂q2i+∂2y∂q1∂q2j+∂2z∂q1∂q2k)=−H1e11H12∂H1∂q1+1H1∂2x∂q1∂q2(e1H1∂x∂q1+e2H2∂x∂q2+e3H3∂x∂q3)+ 1H1∂2y∂q1∂q2(e1H1∂y∂q1+e2H2∂y∂q2+e3H3∂y∂q3)+1H1∂2z∂q1∂q2(e1H1∂z∂q1+e2H2∂z∂q2+e3H3∂z∂q3)$
(A56)

From Eq. A30 we obtain

$∂x∂q1∂2x∂q1∂q2+∂y∂q1∂2y∂q1∂q2+∂z∂q1∂2z∂q1∂q2=12∂H12∂q2=H1∂H1∂q2$
(A57)
$∂x∂q2∂2x∂q1∂q2+∂y∂q2∂2y∂q1∂q2+∂z∂q2∂2z∂q1∂q2=12∂H22∂q1=H2∂H2∂q1$
(A58)

From Eq. A48 we obtain A59A61 when e1e2, and e3 are mutually perpendicular.

$∂x∂q1∂x∂q2+∂y∂q1∂y∂q2+∂z∂q1∂z∂q2=0$
(A59)
$∂x∂q1∂x∂q3+∂y∂q1∂y∂q3+∂z∂q1∂z∂q3=0$
(A60)
$∂x∂q2∂x∂q3+∂y∂q2∂y∂q3+∂z∂q2∂z∂q3=0$
(A61)

Differentiating Eq. A59 with respect to q3, we obtain

$(∂2x∂q1∂q3∂x∂q2+∂2y∂q1∂q3∂y∂q2+∂2z∂q1∂q3∂z∂q2)+(∂x∂q1∂2x∂q2∂q3+∂y∂q1∂2y∂q2∂q3+∂z∂q1∂2z∂q2∂q3) =(X)+(Y)=0$
(A62)

Similarly differentiating Eq. A60 with respect to q2 and differentiating Eq. A61 with respect to q1, we obtain

$(∂2x∂q1∂q2∂x∂q3+∂2y∂q1∂q2∂y∂q3+∂2z∂q1∂q2∂z∂q3)+(∂x∂q1∂2x∂q2∂q3+∂y∂q1∂2y∂q2∂q3+∂z∂q1∂2z∂q2∂q3) =(Z)+(Y)=0$
(A63)
$(∂2x∂q1∂q2∂x∂q3+∂2y∂q1∂q2∂y∂q3+∂2z∂q1∂q2∂z∂q3)+(∂x∂q2∂2x∂q1∂q3+∂y∂q2∂2y∂q1∂q3+∂z∂q2∂2z∂q1∂q3) =(Z)+(X)=0$
(A64)

Equation A65 is the determinant for the three associated Eqs. A62A64. This ensures that X = Y = Z =0. Inserting the results from Eqs. A57 and A60, A62A64 into Eq. A56, we obtain

$∂e1∂q2=−H1e11H12∂H1∂q1+e1H12H1∂H1∂q2+e2H1H2H2∂H2∂q1+0=e2H1∂H2∂q1$
(A65)

The other formulae from A39 to A45 can be proved similarly. From Eq. A48 we obtain Eq. A66 for an orthogonal coordinate system.

$e1=∂rH1∂q1=1H1(∂x∂q1i+∂y∂q1j+∂z∂q1k)e2=∂rH2∂q2=1H2(∂x∂q2i+∂y∂q2j+∂z∂q2k)e3=∂rH3∂q3=1H3(∂x∂q3i+∂y∂q3j+∂z∂q3k)$
(A66)

From Eq. A48, A66 or A50 we obtain Eqs. A67 and A68.

$A=A1e1+A2e2+A2e3=A1∂rH1∂q1+A2∂rH2∂q2+A3∂rH3∂q3=A1H1∇q1+A2H2∇q2+A3H3∇q3=A11H1(∂x∂q1i+∂y∂q1j+∂z∂q1k)+A21H2(∂x∂q2i+∂y∂q2j+∂z∂q2k)+A31H3(∂x∂q3i+∂y∂q3j+∂z∂q3k)=i(A1H1∂x∂q1 + A2H2∂x∂q2 + A3H3∂x∂q3) + j(A1H1∂y∂q1 + A2H2∂y∂q2 + A3H3∂y∂q3) + k(A1H1∂z∂q1 + A2H2∂z∂q2 + A3H3∂z∂q3)=(∑i=13Aiei)⋅[(∑i=13eiHi∂x∂qi)i+(∑i=13eiHi∂y∂qi)j+(∑i=13eiHi∂z∂qi)k]=i∑i=13Ai(i⋅ei)+j∑i=13Ai(j⋅ei)+k∑i=13Ai(k⋅ei)=Axi+Ayj+Azk$
(A67)
$i=e1H1∂x∂q1+e2H2∂x∂q2+e3H3∂x∂q3j=e1H1∂y∂q1+e2H2∂y∂q2+e3H3∂y∂q3k=e1H1∂z∂q1+e2H2∂z∂q2+e3H3∂z∂q3$
(A68)

The relationship between Eqs. A66 and A68 is consistent with Eq. A49.

c. Method 2 for gradient, divergence, and curl in a general and orthogonal coordinate system

From Eq. A68 we obtain Eqs. A69 and A70 for an orthogonal system. Equation A69 is a counterpart of A67.

$A=Axi+Ayj+Azk=Ax(e1H1∂x∂q1+e2H2∂x∂q2+e3H3∂x∂q3)+Ay(e1H1∂y∂q1+e2H2∂y∂q2+e3H3∂y∂q3)+ Az(e1H1∂z∂q1+e2H2∂z∂q2+e3H3∂z∂q3)=e1H1(Ax∂x∂q1+Ay∂y∂q1+Az∂z∂q1)+ e2H2(Ax∂x∂q2+Ay∂y∂q2+Az∂z∂q2)+e3H3(Ax∂x∂q3+Ay∂y∂q3+Az∂z∂q3)=∑i=13eiHi(Ax∂x∂qi+Ay∂y∂qi+Az∂z∂qi)=∑i=13ei{(Axi+Ayj+Azk)⋅[1Hi(∂x∂qii+∂y∂qij+∂z∂qik)]}=∑i=13ei[(Axi+Ayj+Azk)⋅(1Hi∂r∂qi)]=∑i=13ei[(Axi+Ayj+Azk)⋅(Hi∇qi)]=∑i=13ei[(Axi+Ayj+Azk)⋅ei]=A1e1+A2e2+A2e3$
(A69)
$∇f(x,y,z)=i∂f∂x+j∂f∂y+k∂f∂z=∇f[x(q1,q2,q3),y(q1,q2,q3),z(q1,q2,q3)]=∂f∂x(e1H1∂x∂q1+e2H2∂x∂q2+e3H3∂x∂q3)+∂f∂y(e1H1∂y∂q1+e2H2∂y∂q2+e3H3∂y∂q3)+ ∂f∂z(e1H1∂z∂q1+e2H2∂z∂q2+e3H3∂z∂q3)=e1H1(∂f∂x∂x∂q1+∂f∂y∂y∂q1+∂f∂z∂z∂q1)+ e2H2(∂f∂x∂x∂q2+∂f∂y∂y∂q2+∂f∂z∂z∂q2)+e3H3(∂f∂x∂x∂q3+∂f∂y∂y∂q3+∂f∂z∂z∂q3)=e1∂fH1∂q1+e2∂fH2∂q2+e3∂fH3∂q3=∑i=13{eiHi[(i∂f∂x+j∂f∂y+k∂f∂z)⋅(i∂x∂qi+j∂y∂qi+k∂z∂qi)]}=∑i=13{eiHi[(∑j=13ej∂fHj∂qj)⋅∂r∂qi]}=∑i=13{eiHi[∑j=13∂f∂qj(ej⋅∂rHj∂qi)]}=∑i=13(eiHi∂f∂qi)$
(A70)

The final result given by Eq. A70 is the same as that from Eq. A33. From Eqs. A24, A30, and A70, we obtain Eqs. A31 and A71.

$∇f(q1,q2,q3)=∂f∂q1∇q1+∂f∂q2∇q2+∂f∂q3∇q3=e1H1∂f∂q1+e2H2∂f∂q2+e3H3∂f∂q3=e1∂f∂q1∂q1∂s1+e2∂f∂q2∂q2∂s2+e3∂f∂q3∂q3∂s3$
(A71)
$∇⋅A=(e1H1∂∂q1+e2H2∂∂q2+e3H3∂∂q3)⋅(A1e1+A2e2+A3e3)$
(A72)

Using Eqs. A37A45

$e1H1∂∂q1(A1e1)=(e1H1∂A1∂q1⋅e1+e1H1A1⋅∂e1∂q1)=1H1∂A1∂q1+e1H1A1⋅(−e2H2∂H1∂q2−e3H3∂H1∂q3)=1H1∂A1∂q1$
(A73)

All the other terms in Eq. A72 (Table I) can be worked out similarly to complete the derivation. The final result given by Eq. A34 in method 1 can be obtained from Eq. A72 and the results summarized in Table I.

$∇×A=(e1H1∂∂q1+e2H2∂∂q2+e3H3∂∂q3)×(A1e1+A2e2+A3e3)$
(A74)
$e1H1∂∂q1×(A1e1)=e1H1×∂A1∂q1e1+e1H1A1×∂e1∂q1=0+A1e1H1×(−e2H2∂H1∂q2−e3H3∂H1∂q3)=−A1e3H1H2∂H1∂q2+A1e2H1H3∂H1∂q3$
(A75)
TABLE I.

The dot products involved in Eq. A72.

Dot product$e1H1∂∂q1$$e2H2∂∂q2$$e3H3∂∂q3$
$A1e1$ $1H1∂A1∂q1$ $A1H1H2∂H2∂q1$ $A1H1H3∂H3∂q1$
$A2e2$ $A2H1H2∂H1∂q2$ $1H2∂A2∂q2$ $A2H2H3∂H3∂q2$
$A3e3$ $A3H1H3∂H1∂q3$ $A3H2H3∂H2∂q3$ $1H3∂A3∂q3$
Dot product$e1H1∂∂q1$$e2H2∂∂q2$$e3H3∂∂q3$
$A1e1$ $1H1∂A1∂q1$ $A1H1H2∂H2∂q1$ $A1H1H3∂H3∂q1$
$A2e2$ $A2H1H2∂H1∂q2$ $1H2∂A2∂q2$ $A2H2H3∂H3∂q2$
$A3e3$ $A3H1H3∂H1∂q3$ $A3H2H3∂H2∂q3$ $1H3∂A3∂q3$

Finally, Eq. A36 in method 1 is obtained from Eq. A74 using Table II.

TABLE II.

The cross products involved in Eq. A74.

Cross product$e1H1∂∂q1$$e2H2∂∂q2$$e3H3∂∂q3$
$A1e1$ $A1e2H1H3∂H1∂q3−A1e3H1H2∂H1∂q2$ $−e3H2∂A1∂q2$ $e2H3∂A1∂q3$
$A2e2$ $e3H1∂A2∂q1$ $A2e3H1H2∂H2∂q1−A2e1H2H3∂H2∂q3$ $−e1H3∂A2∂q3$
$A3e3$ $−e2H1∂A3∂q1$ $e1H2∂A3∂q2$ $A3e1H2H3∂H3∂q2−A3e2H1H3∂H3∂q1$
Cross product$e1H1∂∂q1$$e2H2∂∂q2$$e3H3∂∂q3$
$A1e1$ $A1e2H1H3∂H1∂q3−A1e3H1H2∂H1∂q2$ $−e3H2∂A1∂q2$ $e2H3∂A1∂q3$
$A2e2$ $e3H1∂A2∂q1$ $A2e3H1H2∂H2∂q1−A2e1H2H3∂H2∂q3$ $−e1H3∂A2∂q3$
$A3e3$ $−e2H1∂A3∂q1$ $e1H2∂A3∂q2$ $A3e1H2H3∂H3∂q2−A3e2H1H3∂H3∂q1$

The only difference between the two methods detailed in Subsections 2a and 2c of the  Appendix is in the process of collecting terms. So essentially the two methods are the same. The first method is more advanced and thus more complex since it uses more mathematical techniques but has the advantage that it is concise and offers an abstract understanding. The second method uses more elementary ways to collect terms. It is lengthy but intuitive thus worthwhile though a little tedious.34

d. Applications to cylindrical and spherical coordinates

To make this Appendix self-contained, we include the applications of the above results in both cylindrical and spherical coordinates.Cylindrical coordinates

q1, q2, and q3 in cylindrical coordinate system are replaced by ρ, φ, and z.19

$x=ρcosφy=ρsinφz=z$
(A76)

From Equation A30 we obtain

$Hρ=(∂x∂ρ)2+(∂y∂ρ)2+(∂z∂ρ)2=1$
(A77)
$Hθ=(∂x∂φ)2+(∂y∂φ)2+(∂zφ)2=ρ$
(A78)
$Hz=(∂x∂z)2+(∂y∂z)2+(∂z∂z)2=1$
(A79)

From Eqs. A32A36 we obtain

$∇f(ρ,φ,z)=eρ∂f∂ ρ+eφρ∂f∂φ+ez∂f∂z$
(A80)
$∇⋅A(ρ,φ,z)=∂(ρAρ)ρ∂ ρ+∂Aφρ∂φ+∂Az∂z$
(A81)
$∇2f(ρ,φ,z)=∂ρ∂ ρ(ρ∂f∂ ρ)+∂2fρ2∂φ2+∂2f∂z2$
(A82)
$∇×A(ρ,φ,z)=1ρeρρeφez∂∂ ρ∂∂φ∂∂zAρρAφAz$
(A83)

Spherical coordinates

q1, q2, and q3 in spherical coordinate system are replaced by r, θ, and φ.

$x=rsinθcosφy=rsinθsinφz=rcosθ$
(A84)

From Eq. A30 we obtain

$Hr=(∂x∂r)2+(∂y∂r)2+(∂z∂r)2=1$
(A85)
$Hθ=(∂x∂θ)2+(∂y∂θ)2+(∂z∂θ)2=r$
(A86)
$Hφ=(∂x∂φ)2+(∂y∂φ)2+(∂z∂φ)2=rsinθ$
(A87)

From Eqs. A32A36 we obtain

$∇f(r,θ,φ)=er∂f∂r+eθr∂f∂θ+eφrsinθ∂f∂φ$
(A88)
$∇⋅A(r,θ,φ)=∂(r2Ar)r2∂r+∂(sinθAθ)rsinθ∂θ+∂Aφrsinθ∂φ$
(A89)
$∇2f(r,θ,φ)=∂r2∂r(r2∂f∂r)+∂r2sinθ∂θ(sinθ∂f∂θ)+∂2fr2sin2θ∂φ2$
(A90)
$∇×A(r,θ,φ)=1r2sinθerreθrsinθeφ∂∂r∂∂θ∂∂φArrAθrsinθAφ$
(A91)

Equations A76A91 are easily available but they are included here for reference.

e. Method 3 for divergence and curl involving reciprocal relationship in a general and orthogonal coordinate system

x, y, and z can be defined with curvilinear coordinates q1, q2, and q3 which can be defined inversely of xyz-space.

$x=x(q1,q2,q3)y=y(q1,q2,q3)z=z(q1,q2,q3)$
(A92)
$q1=q1(x,y,z)q2=q2(x,y,z)q3=q3(x,y,z)$
(A93)

For a surface where qi is a constant, we obtain

$qi=qi(x,y,z)=const. i=1,2,3$
(A94)

The intersection of three such planes identifies a point in the q1q2q3-space.

$dqi=∂qi∂xdx+∂qi∂ydy+∂qi∂zdz=(i∂qi∂x+j∂qi∂y+k∂qi∂z)⋅(idx+jdy+kdz)=[∇qi(x,y,z)]⋅drqi=nqi⋅Tqi=0$
(A95)

The gradient ∇qi is the normal n to the surface of Eq. A94. Three such gradients are associated with the three unit vectors shown for an orthogonal coordinate system in Eqs. A28 and A31. dr in Eq. A95 is the vector T on the tangent plane to the corresponding surface where qi is constant. Two of the surfaces with Eq. A94 define a curve rqi,qj and its tangent Tqi,qj. The intersections of the three surfaces result in three curves. The intersection of these three curves also characterizes a point in the space.

$nqi⋅Tqi,qj=nqj⋅Tqi,qj=0 i≠j$
(A96)

Using Eq. A30 on three of the curves from Eq. A96 we obtain three unit vectors characterized by Eq. A48 or A66 with k = 1, 2, and 3.

$drqi,qj(qk)rqi,qj(qk)=Tqi,qj(qk)Tqi,qj(qk)=i∂x(qk)∂sk+j∂y(qk)∂sk+k∂z(qk)∂sk=[i∂x(qk)∂qk+jdy(qk)dqk+k∂z(qk)∂qk]∂qk∂sk=i∂x(qk)∂qk+jdy(qk)dqk+k∂z(qk)∂qk[∂x(qk)∂qk]2+[dy(qk)dqk]2+[dz(qk)dqk]2=1Hk[i∂x(qk)∂qk+jdy(qk)dqk+k∂z(qk)∂sk]=1Hk∂r(qk)∂qk=ek k≠i,j$
(A97)

Equation A95 represents a system of simultaneous equations with i =1, 2, and 3. From Eq. A95 with Cramer’s rule we obtain

$∂q1∂x=1∂y∂q1∂z∂q10∂y∂q2∂z∂q20∂y∂q3∂z∂q3/∂x∂q1∂y∂q1∂z∂q1∂x∂q2∂y∂q2∂z∂q2∂x∂q3∂y∂q3∂z∂q3=1J1∂y∂q1∂z∂q10∂y∂q2∂z∂q20∂y∂q3∂z∂q3$
(A98)
$∂q1∂y=∂x∂q11∂z∂q1∂x∂q20∂z∂q2∂x∂q30∂z∂q3/∂x∂q1∂y∂q1∂z∂q1∂x∂q2∂y∂q2∂z∂q2∂x∂q3∂y∂q3∂z∂q3=1J∂x∂q11∂z∂q1∂x∂q20∂z∂q2∂x∂q30∂z∂q3$
(A99)
$∂q1∂z=∂x∂q1∂y∂q11∂x∂q2∂y∂q20∂x∂q3∂y∂q30/∂x∂q1∂y∂q1∂z∂q1∂x∂q2∂y∂q2∂z∂q2∂x∂q3∂y∂q3∂z∂q3=1J∂x∂q1∂y∂q11∂x∂q2∂y∂q20∂x∂q3∂y∂q30$
(A100)

Applying Eq. A32 or A71 to q1(x, y, z) we obtain ∇q1. From Eqs. A98A100, we obtain

$∂r(q2)∂q2×∂r(q3)∂q3=ijk∂x∂q2∂y∂q2∂z∂q2∂x∂q3∂y∂q3∂z∂q3=i1∂y∂q1∂z∂q10∂y∂q2∂z∂q20∂y∂q3∂z∂q3−j∂x∂q11∂z∂q1∂x∂q20∂z∂q2∂x∂q30∂z∂q3+∂x∂q1∂y∂q11∂x∂q2∂y∂q20∂x∂q3∂y∂q30k=J(i∂q1∂x+j∂q1∂y+k∂q1∂z)=J∇q1$
(A101)

Equation A101 is consistent with Eqs. A31 and A48 as shown by Eq. A102.

$∂r(q2)∂q2×∂r(q3)∂q3=H2e2×H3e3=H1H2H3H1e1=J∇q1=JH12∂r(q1)∂q1$
(A102)

The other equations in Eq. A31 or A103 can be proved similarly.

$∇q1=∂r(q2)∂q2×∂r(q3)∂q3J=1H1e1∇q2=∂r(q3)∂q3×∂r(q1)∂q1J=1H2e2∇q3=∂r(q1)∂q2×∂r(q2)∂q2J=1H3e3$
(A103)

Equation A103 shows that (∇q1, ∇q2, ∇q3) and $[∂r(q1)∂q2,∂r(q2)∂q2,∂r(q3)∂q3]$ are bases for mutually direct and reciprocal spaces, respectively. The direct and reciprocal spaces are the same for a Cartesian coordinate system and indeed any other orthogonal coordinate system. The reciprocal of Eq. A103 is A104.

$∂r(q1)∂q1=J∇q2×∇q3=H1e1=H12∇q1∂r(q2)∂q2=J∇q3×∇q1=H2e2=H22∇q2∂r(q3)∂q3=J∇q1×∇q2=H3e3=H32∇q3$
(A104)

Equation A104 is consistent with Eqs. A31 and A48 and it can be proved from A105.

$J∇q2×∇q3=J∂r(q3)∂q3×∂r(q1)∂q1J×∂r(q1)∂q1×∂r(q2)∂q2J=1J∂r(q1)∂q1[(∂r(q3)∂q3×∂r(q1)∂q1)⋅∂r(q2)∂q2]−1J[(∂r(q3)∂q3×∂r(q1)∂q1)⋅∂r(q1)∂q1]∂r(q2)∂q2=∂r(q1)∂q1$
(A105)

For an orthogonal coordinate system, Eq. A29 can be rewritten as

$ds2=(∑i=13∂x∂qidqi)2+(∑i=13∂y∂qidqi)2+(∑i=13∂z∂qidqi)2=∑i=13(∂r∂qi2dqi2)+2∑i=13∑j>i3(∂r∂qi⋅∂r∂qjdqidqj)=∑i=13(∂r∂qi2dqi2)$
(A106)

If qj is a constant when j is not equal to i, we obtain a unit vector from Eq. A30.

$∂r∂si=i∂x∂qi∂qi∂si+j∂y∂qi∂qi∂si+k∂z∂qi∂qi∂si=(i∂x∂qi+j∂y∂qi+k∂z∂qi)∂qi∂si=i∂x∂qi+j∂y∂qi+k∂z∂qi(∂x∂qi)2+(∂y∂qi)2+(∂z∂qi)2=ei=1Hi∂r∂qi$
(A107)

Equation A107 is a proof for Eq. A47 and provides the derivation of Eq. A48. We obtain Eq. A108 from Eq. A30 or A107. Equation A108 is consistent with Eq. A48.

$dr=∂r∂q1dq1+∂r∂q2dq2+∂r∂q3dq3=∂(xi+yj+zk)∂q1dq1+∂(xi+yj+zk)∂q2dq2+∂(xi+yj+zk)∂q3dq3=(∂x∂q1i+∂y∂q1j+∂z∂q1k)(∂x∂q1)2+(∂y∂q1)2+(∂z∂q1)2H1dq1+(∂x∂q2i+∂y∂q2j+∂z∂q2k)H2H2dq2+ ∂x∂q3i+∂y∂q3j+∂z∂q3kH3∂q3H3dq3=e1H1dq1+e2H2dq2+e3H3dq3$
(A108)

From Eqs. A67 and A69

$Ax=A⋅i=(A1e1+A2e2+A3e3)⋅i=A1∂rH1∂q1⋅i+A2∂rH2∂q2⋅i+A3∂rH3∂q3⋅i=A1(∂xH1∂q1i+∂yH1∂q1j+∂zH1∂q1k)⋅i+A2(∂xH2∂q2i+∂yH2∂q2j+∂zH2∂q2k)⋅i+ A3(∂xH3∂q3i+∂yH3∂q3j+∂zH3∂q3k)⋅i=A1∂xH1∂q1+A2∂xH2∂q2+A3∂xH3∂q3=A1H1∇q1⋅i+A2H2∇q2⋅i+A3H3∇q3⋅i=A1H1(∂q1∂xi+∂q1∂yj+∂q1∂zk)⋅i+A2H2(∂q2∂xi+∂q2∂yj+∂q2∂zk)⋅i+ A3H3(∂q3∂xi+∂q3∂yj+∂q3∂zk)⋅i=A1H1∂q1∂x+A2H2∂q2∂x+A3H3∂q3∂x$
(A109)
$A1=A⋅∂rH1∂q1=(Axi+Ayj+Azk)⋅(∂xH1∂q1i+∂yH1∂q1j+∂zH1∂q1k)=1H1(Ax∂x∂q1+Ay∂y∂q1+Az∂z∂q1)=A⋅H1∇q1=(Axi+Ayj+Azk)⋅H1(∂q1∂xi+∂q1∂yj+∂q1∂zk)=H1Ax∂q1∂x+Ay∂q1∂y+Az∂q1∂z$
(A110)

From Eq. A109

$A=A1∂rH1∂q1+A2∂rH2∂q2+A3∂rH3∂q3=H2H3A1∂rJ∂q1+H3H1A2∂rJ∂q2+H1H2A3∂rJ∂q3$
(A111)

From Eqs. A5 and A111

$∇⋅A=∇⋅(H2H3A1∂rJ∂q1+H3H1A2∂rJ∂q2+H1H2A3∂rJ∂q3)=[H2H3A1∇⋅∂rJ∂q1+∇(H2H3A1)⋅∂rJ∂q1]+ [H3H1A2∇⋅∂rJ∂q2+∇(H3H1A2)⋅∂rJ∂q2]+ [H1H2A3∇⋅∂rJ∂q3+∇(H1H2A3)⋅∂rJ∂q3]$
(A112)

From Eqs. A3, A7, and A104

$∇⋅∂rJ∂q1=∇⋅(∇q2×∇q3)=∇⋅[∇q2×(∇q3)c]+∇⋅[(∇q2)c×∇q3](∇×∇q2)⋅∇q3−(∇×∇q3)⋅∇q2=0$
(A113)

Equation A112 can be reduced by the result from Eqs. A32 and A113.

$∇⋅A=H1J∇(H2H3A1)⋅∂rH1∂q1+H2J∇(H3H1A2)⋅∂rH2∂q2+H3J∇(H1H2A3)⋅∂rH3∂q3=H1J∇(H2H3A1)⋅e1+H2J∇(H3H1A2)⋅e2+H3J∇(H1H2A3)⋅e3=H1J(e1∂H1∂q1+e2∂H2∂q2+e3∂H3∂q2)(H2H3A1)⋅e1+1J∂(H3H1A2)∂q2+1J∂(H1H2A3)∂q2$
(A114)

The result of Eq. A114 is the same as that obtained from Eq. A34. From Eqs. A6, A104, and A111, we obtain Eq. A115.

$∇×A=∇×(H1A1∂rH12∂q1+H2A2∂rH22∂q2+H3A3∂rH32∂q3)=∇(H1A1)×∂rH12∂q1+H1A1∇×∂rH12∂q1+ ∇(H2A2)×∂rH22∂q2+H2A2∇×∂rH22∂q2+ ∇(H3A3)×∂rH32∂q3+H3A3∇×∂rH32∂q3$
(A115)

From Eqs. A3 and A104

$∇×∂rH12∂q1=∇×∇q1=0$
(A116)

Using the results from Eqs. A103 and A116, the e1 component of Eq. A115 can be obtained.

$(∇×A)⋅∂rH1∂q1=[∇(H1A1)×∂rH12∂q1]⋅∂rH1∂q1++ [∇(H2A2)×∂rH22∂q2]⋅∂rH1∂q1+[∇(H3A3)×∂rH32∂q3]⋅∂rH1∂q1=0+∇(H2A2)⋅(∂rH22∂q2×∂rH1∂q1)+∇(H3A3)⋅(∂rH32∂q3×∂rH1∂q1)=e3∂(H2A2)H3∂q3⋅(−JH22H1e3H3)+e2∂(H3A3)H2∂q2⋅JH32H1e2H2=1H2H3[∂(H3A3)∂q2−∂(H2A2)∂q3]$
(A117)

It can be proved from Eq. A117 that the result from Eq. A115 is the same as that from Eq. A36. The method introduced to calculate ∇⋅A and ∇×A in this section is essentially the same as that introduced in Subsection 2a of the  Appendix. However, the method is more advanced since the derivations are helped by the concepts of reciprocal vectors and much work is needed to develop these concepts. Every method, simple or involved, has its own merits.34

f. Proofs for Cramer’s rule

Cramer’s rule has been used to Eq. A95 to obtain Eqs. A98A100. From Eq. A95 the derivatives of q1 with respect to the three independent variables q1, q2 and q3 are

$∂q1∂x∂x∂q1+∂q1∂y∂y∂q1+∂q1∂z∂z∂q1=1$
(A118)
$∂q1∂x∂x∂q2+∂q1∂y∂y∂q2+∂q1∂z∂z∂q2=0$
(A119)
$∂q1∂x∂x∂q3+∂q1∂y∂y∂q3+∂q1∂z∂z∂q3=0$
(A120)

Proof 1

Multiply column 1 of determinant J by ∂q1/∂x resulting in Eq. A121 from which (A98) is obtained.

$J∂q1∂x=∂x∂q1∂q1∂x∂y∂q1∂z∂q1∂x∂q2∂q1∂x∂y∂q2∂z∂q2∂x∂q3∂q1∂x∂y∂q3∂z∂q3=∂x∂q1∂q1∂x+∂y∂q1∂q1∂y+∂z∂q1∂q1∂z∂y∂q1∂z∂q1∂x∂q2∂q1∂x+∂y∂q2∂q1∂y+∂z∂q2∂q1∂z∂y∂q2∂z∂q2∂x∂q3∂q1∂x+∂y∂q3∂q1∂y+∂z∂q3∂q1∂z∂y∂q3∂z∂q3=1∂y∂q1∂z∂q10∂y∂q2∂z∂q20∂y∂q3∂z∂q3$
(A121)

Proof 2

Jij is the cofactor of the element at row i and column j in the determinant J. Equation A99 can be obtained from A123.

$(A118)×J12+(A119)×J22+(A120)×J32$
(A122)
$∂y∂q1∂q1∂y×J12+∂y∂q2∂q1∂y×J22+∂y∂q3∂q1∂y×J32=J∂q1∂y=1×J12+0×J22+0×J32=∂x∂q11∂z∂q1∂x∂q20∂z∂q2∂x∂q30∂z∂q3$
(A123)

Proof 3

Equations A118A120 can be written as a matrix form and Eqs. A98A100 can be obtained from Eq. A125.

$∂x∂q1∂y∂q1∂z∂q1∂x∂q2∂y∂q2∂z∂q2∂x∂q3∂y∂q3∂z∂q3∂q1∂x∂q1∂y∂q1∂z=100$
(A124)
$∂q1∂x∂q1∂y∂q1∂z=∂x∂q1∂y∂q1∂z∂q1∂x∂q2∂y∂q2∂z∂q2∂x∂q3∂y∂q3∂z∂q3−1100=1JJ11J21J31J12J22J32J13J23J33100=1JJ11×1+J21×0+J31×0J12×1+J22×0+J32×0J13×1+J23×0+J33×0$
(A125)
g. Green’s, Gauss’s, and Stokes’s theorems

Applications are given to reveal the physical meaning behind the mathematical derivations for the three methods introduced in Subsections 1, 3, and 5 of the  Appendix respectively. Gauss’s, and Stokes’s theorems35 have been related to Green’s theorem. f(q1, q2, q3) is a continuous scalar function defined in an orthogonal coordinate system. The flux at point P(q1, q2, q3) characterized by f is

$∇f=e1∂fH1∂q1+e2∂fH2∂q2+e3∂fH3∂q3$
(A126)

Gauss’s theorem (also known as the divergence theorem)

Point P is enclosed by a volume dV1. The volume is bounded by surface S1.

$dV1=∂r∂q1dq1⋅(∂r∂q2dq2×∂r∂q3dq3)=H1dq1e1⋅(H2dq2e2×H3dq3e3)=H1H2H3dq1dq2dq3=∂x∂q1∂y∂q1∂z∂q1∂x∂q2∂y∂q2∂z∂q2∂x∂q3∂y∂q3∂z∂q3dq1dq2dq3=Jdq1dq2dq3$
(A127)

 in Eq. A127 implies the absolute value of a determinant. The flux moving out of the volume V1 or surface S1 is given by

$∂∂q1[(∂fH1∂q1)H2dq2H3dq3]dq1+∂∂q2(H1H3H2∂f∂q2)dq1dq2dq3+∂∂q3(H1H2H3∂f∂q2)dq1dq2dq3$
(A128)

If volume V is accumulated from V1, V2, V3, …, VN, and S is the surface of the volume V, the flux moved out of the volume V or the surface S is given by the divergence theorem19,32,33 Eq. A129 with reference to Eq. A35.

$∮S∇f⋅dS=∮S[(e1H1∂f∂q1)+(e2H2∂f∂q2)+(e3H3∂f∂q3)]⋅e1H2H3dq2dq3+ e2H1H3dq1dq3+e3H1H2dq1dq2=∫V1H1H2H3∂∂q1(H2H3H1∂f∂q1)+∂∂q2(H1H3H2∂f∂q2)+ ∂∂q3(H1H2H3∂f∂q3)H1H2H3dq1dq2dq=∫V[∇⋅∇f(q1,q2,q3)]dV=∫V[∇2f(q1,q2,q3)]dV$
(A129)

The flux from the interface between adjacent volumes Vi and Vj is zero as the net flux leaving one cancels the net flux into the other. The net flux is accumulated on S. Equation A129 is also consistent with Eqs. A34 and A130. n in Eq. A130 is the unit vector normal to surface S.

$∮SA(q1,q2,q3)⋅dS=∮SA(q1,q2,q3)⋅ndS=∮SA1e1⋅(e2H2dq2×e3H3dq3)+ A2e2⋅(e3H3dq3×e1H1dq1)+A3e3⋅(e1H1dq1×e2H2dq2)=∭∂(A1H2H3)∂q1dq1dq2dq3+∂(A2H1H3)∂q2dq1dq2dq3+ ∂(A3H1H2)∂q3dq1dq2dq3=∫V(∇×A)H1H2H3dq1dq2dq=∫V∫V(∇×A)dV$
(A130)

When the dimension is reduced by one in Eq. A130, i.e. S and V are reduced to a closed curve C and area S, respectively. The normal of S is parallel to e3. Then, a form of Green’s theorem, Eq. A131, is obtained. n is the unit vector perpendicular to curve C in the two-dimensional plane.

$∮CA(q1,q2)⋅ndC(q1,q2)=∮S(A1e1+A2e2)⋅(e1H2dq2−e2H1dq1)=∮S(−A2H1dq1+A1H2dq2)=∬S{e3⋅[e1∂(A1H2dq2)∂q1dq1×e2]−e3⋅[e2∂(A2H1dq1)∂q2dq2×e1]}=∬S{1H1H2[∂(A1H2)∂q1]+[∂(A2H1)∂q2]}H1H2dq1dq2=∬S∇⋅AH1H2dq1dq2=∬S∇⋅AdS$
(A131)

Stokes’s theorem

Vector A is defined in a three-dimensional space. C is a closed curve and S is the two-dimensional surface enclosed by C. The normal of S is aligned with e3. There is a form of Green’s theorem, shown by Eq. A132.19T is the unit vector tangent to curve C.

$∮CA(q1,q2)⋅TdC(q1,q2)=∮C(e1A1+e2A1)⋅(e1H1dq1+e2H2dq2)=∮C(A1H1dq1+A2H2dq2)=∬S∂(A2H2dq2)H1∂q1e3⋅(H1dq1e1×e2)+∂(A1H1dq1)H2∂q2e3⋅ (H2dq2e2×e1)=∬S[(∂(A2H2)∂q1−∂(A1H1)∂q2)e3]⋅(e1H1dq1×e2H2dq2)H1H2=∬S{1H1H2[∂(A2H2)∂q1−∂(A1H1)∂q2]e3}⋅dS=∬S(∇×A)⋅dS=∬S(∇×A)⋅e3dS$
(A132)

Equations A131 and A132 are consistent.19 Green’s theorem shown by Eq. A132 is relevant for a two-dimensional surface specially orientated in the three-dimensional space. Equation A133 shows how Green’s theorem evolves from a generally orientated two-dimensional surface to Stokes’s theorem in three-dimensional space, where the normal of S is orientated generally in an orthogonal three-dimensional coordinate system. This resolution has been rarely shown in the literature.

C1 is a closed curve in this three-dimensional space. S1 is the surface enclosed by C1. The circulation of A is the line integral of the vector A along the closed path C1. Applying Green’s theorem in the three axial directions and using Eq. A36 we obtain

$∮C1A⋅dC1=∮C1(e1A1+e2A2+e3A3)⋅(e1H1dq1+e2H2dq2+e3H3dq3)=∮C1(A1H1dq1+A2H2dq2+A3H3dq3)=∮C1e3⋅[e1∂(A2H2dq2)H1∂q1H1dq1×e2+e2∂(A1H1dq1)H2∂q2H2dq2×e1]+ e2⋅[e1∂(A3H3dq3)H1∂q1H1dq1×e3+e3∂(A1H1dq1)H3∂q3H3dq3×e1]+ e1⋅[e2∂(A3H3dq3)H2∂q2H2dq2×e3+e3∂(A2H2dq2)H3∂q3H3dq3×e2]=∮C1[∂(A2H2)∂q1e3H1H2⋅(e1H1dq1×dq2H2e2)+∂(A1H1)∂q2e3H1H2]⋅(e2H2dq2×e1H1dq1)+ [(−∂(A3H3)∂q1+∂(A1H1)∂q3)e2H1H3]⋅(H1e1dq1×H3e3dq3)+ 1H2H3[∂(A3H3)∂q2−∂(A2H2)∂q3]H2dq2H3dq3=∬S1(∇×A)⋅ndS1=∬S1(∇×A)⋅dS1$
(A133)

If an open surface S is bounded by a closed contour C, i.e. the surface S can be divided into elementary surfaces S1, S2, S3, …, SN bounded by closed paths C1, C2, C3, …, CN, respectively, then Stokes’s theorem19,32,33 is obtained from Eq. A36.

$∮CA⋅dC=∫S1H1H2[∂(A2H2)∂q1−∂(A1H1)∂q2]H1H2dq1dq2+ 1H1H3[∂(A1H1)∂q3−∂(A3H3)∂q1]H1H3dq1dq3+ 1H2H3[∂(A3H3)∂q2−∂(A2H2)∂q3]H2dq2H3dq3=∫S(∇×A)⋅dS$
(A134)

3. Elliptical coordinates

This short section is an extension related with Subsection 2 of the  Appendix.

a. Elliptical coordinate system

An elliptical coordinate system is also an orthogonal coordinate system. In the metric tensor form and for orthogonal coordinate system with Eq. A48, we obtain

$∂x∂qi∂x∂qj+∂y∂qi∂y∂qj+∂z∂qi∂z∂qj=∂r∂qi⋅∂r∂qj=Hi2; i=j0; i≠j$
(A135)
$(dV)2=∂x∂q1∂y∂q1∂z∂q1∂x∂q2∂y∂q2∂z∂q2∂x∂q3∂y∂q3∂z∂q3∂x∂q1∂x∂q2∂x∂q3∂y∂q1∂y∂q2∂y∂q3∂z∂q1∂z∂q2∂z∂q3(dq1dq2dq3)2=H12000H22000H32(dq1dq2dq3)2=H12H22H32(dq1dq2dq3)2=∂x∂q1∂x∂q2∂x∂q3∂y∂q1∂y∂q2∂y∂q3∂z∂q1∂z∂q2∂z∂q3∂x∂q1∂y∂q1∂z∂q1∂x∂q2∂y∂q2∂z∂q2∂x∂q3∂y∂q3∂z∂q3(dq1dq2dq3)2=(dxdydz)2$
(A136)

 is unnecessary in Eq. A136 since the determinant is already positive.34

The elliptic coordinates ξ, ζ, φ36 are defined by

$ξ=ra+rbR; 1≤ξ≤∞$
(A137)
$ζ=ra−rbR; −1≤ζ≤1$
(A138)
$0≤φ≤2π$
(A139)

From Fig. 6 we obtain

$OD2=r2sin2θ=ra2−(R2+z)2=rb2−(R2−z)2$
(A140)
FIG. 6.

Elliptic coordinates for the hydrogen molecular ion $H2+$. OD is in the xy plane. The distance between the two hydrogen nuclei at points a and b is R. The electron located at P is characterized by r. |r| = r, |ra| = ra, |rb| = rb.

FIG. 6.

Elliptic coordinates for the hydrogen molecular ion $H2+$. OD is in the xy plane. The distance between the two hydrogen nuclei at points a and b is R. The electron located at P is characterized by r. |r| = r, |ra| = ra, |rb| = rb.

Close modal

Solving Eq. A140 we obtain

$z=ra2−rb22R=(ra−rb)(rb+ra)2R=R2ξζ$
(A141)
$OD2=R24(ξ−ζ)2−R24(1+ξζ)2=R24(ξ2−1)(1−ζ2)$
(A142)

Thus

$x=R2(ξ2−1)1/2(1−ζ2)1/2cosφ$
(A143)
$y=R2(ξ2−1)1/2(1−ζ2)1/2sinφ$
(A144)
$dV=R2(1−ζ2)1/2(ξ2−1)−1/2ξcosφR2(1−ζ2)1/2(ξ2−1)−1/2ξsinφR2ζ−R2(ξ2−1)1/2(1−ζ2)−1/2ζcosφ−R2(ξ2−1)1/2(1−ζ2)−1/2ζsinφR2ξ−R2(ξ2−1)1/2(1−ζ2)1/2sinφR2(ξ2−1)1/2(1−ζ2)1/2cosφ0dξdζdφ=[R38ζ2(ξ2−1)−cosφ−sinφ−sinφcosφ−R38ξ2(1−ζ2)cosφsinφ−sinφcosφ]dξdζdφ=H1H2H3dξdζdφ=R38(ξ2−ζ2)dξdζdφ$
(A145)

Since ξζ, the | | symbol is unnecessary outside (ξ2 - ζ2) in Eq. A145.

$Hξ=(∂x∂ξ)2+(∂y∂ξ)2+(∂z∂ξ)2=R2(ξ2−ζ2)12(ξ2−12)12$
(A146)
$Hζ=(∂x∂ζ)2+(∂y∂ζ)2+(∂z∂ζ)2=R2(ξ2−ζ2)12(12−ζ2)12$
(A147)
$Hφ=(∂x∂φ)2+(∂y∂φ)2+(∂z∂φ)2=R2(ξ2−1)1/2(1−ζ2)1/2$
(A148)
$∇2f(r,θ,φ)=4R2(ξ2−ζ2)∂∂ξ[(ξ2−1)∂f∂ξ]+∂∂ζ[(1−ζ2)∂f∂ζ]+ ξ2−ζ2(ξ2−1)(1−ζ2)∂2f∂φ2$
(A149)
b. An application of elliptical coordinates

The overlap integral Sab of 1s orbital in a hydrogen molecular ion is given by:

$Sab=∫Vϕa(1s)ϕb(1s)dV=R38π∫Ve−rae−rb(ξ2−ζ2)dξdζdφ=R38π∫Ve−Rξξ2dξdζdφ−R38π∫Ve−Rξζ2dξdζdφ=4πR38π[(−e−Rξξ2R1∞+2R∫1∞e−Rξξdξ)−13∫1∞e−Rξdξ]=R32(e−RR−2e−RξξR21∞+2R2∫1∞e−Rξdξ−e−R3R)=R32(e−RR+2e−RR2+2e−RR3−e−R3R)=e−R(1+R+R23)$
(A150)

4. Left and right hand rules

The following facts can be helpful when considering whether the right R or the left L hand rules shown in Fig. 8 should be used. The movement of a closed conducting loop shown in Fig. 7a will increase the magnetic flux passing through the loop. An electric current I with magnetic moment μ will be induced in the loop to counteract this increase. The effect of the induced current is always to counteract the flux change in the loop and from it we know that the right hand R rule shown in Fig 7a should be used to predict the direction of current I relative to that of the velocity υ and magnetic field B. In the right hand R rule, the magnetic lines are directed perpendicular to the palm and when the thumb is in the direction of υ, then the four fingers, which are perpendicular to both, point to the direction of I.19

FIG. 7.

The right R (a, b) and left L (c, d) hand rules in electromagnetism. υ indicates the direction and velocity of the movement of a closed loop in the magnetic field B. μ is the magnetic moment of the loop. I is the current. N and S are magnetic poles.

FIG. 7.

The right R (a, b) and left L (c, d) hand rules in electromagnetism. υ indicates the direction and velocity of the movement of a closed loop in the magnetic field B. μ is the magnetic moment of the loop. I is the current. N and S are magnetic poles.

Close modal

The current I in the loop in Fig. 7c generates a magnetic moment μ. The moment will be aligned with the magnetic field B which induces a movement of the loop. This fact can be used to deduce that the left hand L rule shown in Fig 7c should be used to predict the movement of the loop with relation to the current and magnetic field. In the left hand L rule, the magnetic lines are perpendicular to the palm and the four fingers are in the direction of I, then the thumb is in the direction of movement indicated by υ.19

Furthermore, a simple mnemonic37 can be used to remember whether the R or L hand rule should be used. Fig. 7a is related to the electric generator in which a movement of a loop generates electricity. As indicated in Fig. 7b the stroke for “m” in “move” is similar to that for R in “right”, thus as a mnemonic the right hand R should be used in this situation. Similarly, Fig. 7c is related to an electric motor where electricity induces movement. As indicated in Fig. 7d the stroke for “e” in “electricity” is similar to that for L in “left”, thus as a mnemonic the left hand L should be used here.

An electric loop perpendicular to the paper in Fig. 8 is in an inhomogeneous magnetic field. The magnetic moment μ of the loop is aligned with the magnetic field B. Thus, the loop will move upward toward the stronger field to increase the magnetic flux passing through the loop thus reducing energy. This fact is consistent with the left hand L rule shown in Fig. 2a in predicting the direction of the induced force. The force component $F∥$ in the plane along the loop has been canceled because of symmetry. The resultant component $F⊥$ moves the loop upward.

FIG. 8.

The upward (a) and downward (b) movements of current loops in an inhomogeneous magnetic field. Magnetic lines are emitted from point B. F is the force induced by the current loop in the magnetic field. The symbols used are the same as those in Fig. 7.

FIG. 8.

The upward (a) and downward (b) movements of current loops in an inhomogeneous magnetic field. Magnetic lines are emitted from point B. F is the force induced by the current loop in the magnetic field. The symbols used are the same as those in Fig. 7.

Close modal

The magnetic moment μ of the electric loop in Fig. 8b is in the opposite direction to the magnetic field B. Thus, the loop will move downward not allowing the magnetic moment of the loop to reduce the magnetic flux passing through the loop and the energy of the system is reduced. This result can also be predicted by the left hand L rule. Again, the force component $F∥$ in the plane along the loop has been canceled and the resultant component $F⊥$ moves the loop downward.

The discovery of electron spins from the Stern-Gerlach experiment carried out in 1922 was related with the principle implied in Fig. 8. The result of the experiment was not understood from fundamental principles until several years later in 1925, when an entirely new perspective extended the concept to nuclear spins and Zeem Effect in quantum mechanics. It shows that review from what is already known is also originative.

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