# Appendix C: Mathematical Formulas

# Trigonometric Relations

- sin(x±y)=sin x cos y±cos x sin y

- cos(x±y)=cos x cos y∓sin x sin y

- 2 sin x sin y=cos (x−y)−cos (x+y)

- 2 sin x cos y=sin (x+y)+sin (x−y)

- 2 cos x cos y=cos (x+y)+cos (x−y)

- sin 2 x=2 sin x cos x

- cos 2 x=1−2 sin 2 x

- sin x+sin y=2 sin x+y 2 cos x−y 2

- sin x−sin y=2 cos (x+y 2) sin (x−y 2)

- cos x+cos y=2 cos x+y 2 cos x−y 2

- cos x−cos y=−2 sin x+y 2 sin x−y 2

- cos(x±90°)=∓sin x

- cos(−x)=cos x

- sin(x±90°)=±cos x

- sin(−x)=−sin x

- e j x=cos x+j sin x(Euler’s identity)

- sin x=e j x−e−j x 2 j

- cos x=e j x+e−j x 2

# Approximations for Small Quantities

For |x|≪1,

- (1±x)n≈1±n x

- (1±x)2≈1±2 x

- 1±x≈1±x 2

- 1 1±x≈1∓x 2

- e x=1+x+x 2 2!+⋯≈1+x

- ln(1+x)≈x

- sin x=x−x 3 3!+x 5 5!+⋯≈x

- cos x=1−x 2 2!+x 4 4!+⋯≈1−x 2 2

- lim x → 0 sin x x=1

# Common Derivatives

C.3 1 Full Alternative Text

# Common Integrals

- ∫x n d x=1 n+1 x n+1+C

- ∫1 x d x=ln|x|+C

- ∫x−n d x=1−n+1 x−n+1+C

- ∫1 a x+b d x=1 a ln|a x+b|+C

- ∫cos (a x)d x=1 a sin (a x)+C

- ∫sin (a x)d x=−1 a cos (a x)+C

- ∫tan(a x)d x=1 a ln|sec(a x)|+C

- ∫e a x d x=1 a e a x+C

- ∫b x d x=b x ln(b)+C

- ∫​ln x d x=x ln(x)−x+C

- ∫e a x sin (b x)d x=e a x a 2+b 2(a sin (b x)−b cos (b x))+C

- ∫e a x cos (b x)d x=e a x a 2+b 2(a cos (b x)+b sin (b x))+C

- ∫x e x d x=(x−1)e x+C

# Gradient, Divergence, Curl, and Laplacian Operators

# Cartesian (Rectangular) Coordinates (x, y, z)

∇V=x ˆ∂V∂x+y ˆ∂V∂y+z ˆ∂V∂z∇⋅A=∂A x∂x+∂A y∂y+∂A z∂z∇× A =x ˆ y ˆ z ˆ∂∂x∂∂y∂∂z A x A y A z=x ˆ ∂A z∂y−∂A y∂z +y ˆ ∂A x∂z−∂A z∂x +z ˆ ∂A y∂x−∂A x∂y∇2 V=∂2 V∂x 2+∂2 V∂y 2+∂2 V∂z 2

# Cylindrical Coordinates (r, ϕ, z)

∇V=r ˆ∂V∂r+ϕ ˆ 1 r∂V∂ϕ+z ˆ∂V∂z∇⋅A=1 r∂∂r(r A r)+1 r∂A ϕ∂ϕ+∂A z∂z∇× A=1 r r ˆ ϕ ˆ r z ˆ∂∂r∂∂ϕ∂∂z A r r A ϕ A z=r ˆ 1 r∂A z∂ϕ−∂A ϕ∂z +ϕ ˆ∂A r∂z−∂A z∂r +z ˆ 1 r∂∂r(r A ϕ)−∂A r∂ϕ∇2 V=1 r∂∂r r∂V∂r+1 r 2∂2 V∂ϕ 2+∂2 V∂z 2

# Spherical Coordinates (R, θ, ϕ)

∇V=R ˆ∂V∂R+θ ˆ 1 R∂V∂θ+ϕ ˆ 1 R sin θ∂V∂ϕ∇⋅A=1 R 2∂∂R(R 2 A R) +1 R sin θ∂∂θ(A θ sin θ) +1 R sin θ∂A ϕ∂θ∇× A=1 R 2 sin θ R ˆ θ ˆ R ϕ ˆ R sin θ∂∂R∂∂θ∂∂ϕ A R R A θ(R sin θ)A ϕ=R ˆ 1 R sin θ∂∂θ(A ϕ sin θ)−∂A θ∂ϕ +θ ˆ 1 R 1 sin θ∂A R∂ϕ−∂∂R(R A ϕ) +ϕ ˆ 1 R∂∂R(R A θ)−∂A R∂θ∇2 V=1 R 2∂∂R R 2∂V∂R +1 R 2 sin θ∂∂θ sin θ∂V∂θ +1 R 2 sin 2 θ∂2 V∂ϕ 2

# Some Useful Vector Identities

- A⋅B=A B cos θ A B Scalar (or dot) product

- A×​ B=n ˆ A B sin θ A B Vector (or cross) product, n ˆ normal to plane containing A and B

- A⋅(B×​C)=B⋅(C×​A)=C⋅(A×​B)

- A×​(B×C)=B(A⋅C)−C(A⋅B)

- ∇(U+V)=∇U+∇V

- ∇(U V)=U∇V+V∇U

- ∇⋅(A+B)=∇⋅A+∇⋅B

- ∇⋅(U A)=U∇⋅A+A⋅∇U

- ∇×​(U A)=U∇×​ A+∇U×​ A

- ∇×​(A+B)=∇×A+∇×​ B

- ∇⋅(A×B)=B⋅(∇×​A)−A⋅(∇×​ B)

- ∇⋅(∇×A)=0

- ∇×∇​ V=0

- ∇⋅∇V=∇2 V

- ∇×∇×A=∇(∇⋅A)−∇2 A

- ∫υ(∇⋅A) d υ=∮S A⋅d s Divergence theorem (S encloses υ)

- ∫S(∇×​A)⋅d s=∮C A⋅d l Stoke's theorem (S bounded by C)