Electricity & Magnetism I

Northeastern Illinois University

Electricity & Magnetism I Vector Calculus

Greg Anderson Department of Physics & Astronomy Northeastern Illinois University

Spring 2017

c 2004-2017 G. Anderson Electricity & Magnetism – slide 1 / 68 Northeastern Illinois Overview University

Maxwell’s Equations Vector Algebra Rotation of Vectors Grad & Div Curl & Stokes’ Thm. More Vector Calculus Curvilinear Coordinates

c 2004-2017 G. Anderson Electricity & Magnetism – slide 2 / 68 Northeastern Illinois University

Maxwell’s Equations Electromagnetism Maxwell’s Equations In Pictures Selected E&M Heros

Vector Algebra Rotation of Vectors Maxwell’s Equations Grad & Div Curl & Stokes’ Thm. More Vector Calculus Curvilinear Coordinates

c 2004-2017 G. Anderson Electricity & Magnetism – slide 3 / 68 Northeastern Illinois Electromagnetism University

With the exception gravity, almost every force that you experience in everyday life is electro-magnetic in origin. • EM forces bind electrons and nuclei into atoms.

• EM forces bind atoms into molecules. O H H • EM forces bind atoms & molecules into solids.

Electric forces are produced by electric charges.

c 2004-2017 G. Anderson Electricity & Magnetism – slide 4 / 68 Northeastern Illinois Maxwell’s Equations in Vacuum University

Diﬀerential form:

1 ∇· E = ǫ0 ρ, ∇· B = 0

∂B ∂E ∇× E = − ∂t , ∇× B = µ0 J + ǫ0 ∂t Integral form:

E · dA = 1 d3xρ(x), B · dA = 0 S ǫ0 V S H R H ∂ ∂E C E · dℓ = − ∂t B · dA, C B · dℓ = µ0 S J + ǫ0 ∂t · dA H R H R

c 2004-2017 G. Anderson Electricity & Magnetism – slide 5 / 68 Northeastern Illinois Maxwell’s Equations in Vacuum University

Gauss’s Law No magnetic monopoles:

1 ΦE = ǫ Q 0 ΦB =0

Faraday’s Law: Amp´ere-Maxwell Law:

e.g., I, increasing E

E B

e.g., increasing B Φ Φ E s d B B s d E E = H · d = − dt H · d = µ0(I + ǫ0 dt ) c 2004-2017 G. Anderson Electricity & Magnetism – slide 6 / 68 Northeastern Illinois Selected E&M Heros University

Volta Amp´ere Oersted Coulomb Stokes Laplace Maxwell Galvani Thomson (Kelvin) Franklin Faraday DuFey Gauss 1700 1750 1800 1850 1900 1950

c 2004-2017 G. Anderson Electricity & Magnetism – slide 7 / 68 Northeastern Illinois University

Maxwell’s Equations

Vector Algebra Outline Scalars & Vectors Vectors Vector Addition Vector Addition Active Learning Vector Vector Algebra Subtraction Vector Multiplication Dot Product Dot Product II Active Learning Determinants Cross Product Anti- Commutative Levi-Civita Tensor Active Learning A · (B × C) Kronecker Delta Vec. Triple Prod. Rotation of Vectors c 2004-2017 G. Anderson Electricity & Magnetism – slide 8 / 68 Grad & Div Northeastern Illinois Outline University

• Vector Algebra – Deﬁnitions – Addition & Multiplication – Vector Product Identities – Geometric Interpretation

• Transformations of Vectors • Vector Diﬀerential Operators

c 2004-2017 G. Anderson Electricity & Magnetism – slide 9 / 68 Northeastern Illinois Scalars & Vectors University

• A scalar is a quantity which is characterized by its magnitude, e.g., charge, density, energy. • A vector is a quantity with a magnitude and direction, e.g., position, electric & magnetic ﬁelds. z In terms of components:

A ˆ A ,A ,A . k y A =( x y z) ˆi ˆj In terms of unit vectors:

x A = Axˆi + Ayˆj + Azkˆ

c 2004-2017 G. Anderson Electricity & Magnetism – slide 10 / 68 Northeastern Illinois Vectors University z Position vector: x ˆ k y x = xˆi + yˆj + zkˆ ˆi ˆj Magnitude: |x|= x2 + y2 + z2 =r x p

Unit vectors: | ˆi |=| ˆj |=| kˆ |= 1. Equivalent notation:

ˆi = eˆx = eˆ1 ˆj = eˆy = eˆ2 kˆ = eˆz = eˆ3

c 2004-2017 G. Anderson Electricity & Magnetism – slide 11 / 68 Northeastern Illinois Vector Addition University

Vector addition is commutative: A + B = B + A Vector addition is associative: A +(B + C)=(A + B)+ C

c 2004-2017 G. Anderson Electricity & Magnetism – slide 12 / 68 Northeastern Illinois Vector Addition University

Analytic Method Graphical Method Vectors add head to tail. To sum two vectors, sum their If C = A + B, then: components. If C = A + B, then:

Cx = Ax + Bx Cy = Ay + By Cz = Az + Bz C B

A = Axî + Ayˆj + Azkˆ B = Bxî + Byˆj + Bzkˆ A C = Cxî + Cyˆj + Czkˆ

c 2004-2017 G. Anderson Electricity & Magnetism – slide 13 / 68 Northeastern Illinois Active Learning: Vector Addition University

Consider a triangle ABC Which of the following is true? a) A = B − C b) A = C − B c) A = −B − C C B d) None of the Above e) Depends on where the origin A of the coord system is.

c 2004-2017 G. Anderson Electricity & Magnetism – slide 14 / 68 Northeastern Illinois Active Learning: Vector Addition University

Consider a triangle ABC Which of the following is true? a) A = B − C b) A = C − B c) A = −B − C C B d) None of the Above e) Depends on where the origin A of the coord system is.

c 2004-2017 G. Anderson Electricity & Magnetism – slide 14 / 68 Northeastern Illinois Vector Subtraction University

Minus A (−A) is a vector with To subtract a vector, add its op- the same magnitude but opposite posite: direction than A. A − B = A +(−B)

−A B − B A

A = Axˆi + Ayˆj + Azkˆ A −A = −Axˆi − Ayˆj − Azkˆ

c 2004-2017 G. Anderson Electricity & Magnetism – slide 15 / 68 Northeastern Illinois Multiplication with Vectors University

Multiplication of Scalars and Vectors

vector times scalar = vector

cA = c(Ax,Ay,Az)=(cAx, cAy, cAz)= Ac

Multiplication of two vectors

vector · vector = scalar Scalar (dot) product

vector × vector = vector Vector (cross) product

c 2004-2017 G. Anderson Electricity & Magnetism – slide 16 / 68 Northeastern Illinois The Vector Dot Product University

Dot Product, aka Scalar Product, aka Inner Product.

A · B = AxBx + AyBy + AzBz 3 = i=1 AiBi P = AiBi (Einstein summation convention)

B Geometrical Interpretation A · B = | A || B | cos θ = | A | Bk = A | B | A k θ cos θ = A · B/AB

c 2004-2017 G. Anderson Electricity & Magnetism – slide 17 / 68 Northeastern Illinois Properties of the Dot Product University

The dot product is commutative A · B = B · A

The dot product is distributive A · (B + C)= A · B + A · C

c 2004-2017 G. Anderson Electricity & Magnetism – slide 18 / 68 Northeastern Illinois Active Learning University

Using the deﬁnition of the scalar product:

A · B = AxBx + AyBy + AzBz,

show that the dot product is distributive.

c 2004-2017 G. Anderson Electricity & Magnetism – slide 19 / 68 Northeastern Illinois Active Learning University

Using the deﬁnition of the scalar product:

A · B = AxBx + AyBy + AzBz,

show that the dot product is distributive.

A · (B + C) = Ax(Bx + Cx)+ Ay(By + Cy)+ Az(Bz + Cz)

= AxBx + AyBy + AzBz + AxCx + AyCy + AzCz

= A · B + A · C

c 2004-2017 G. Anderson Electricity & Magnetism – slide 19 / 68 Northeastern Illinois Determinants University For a 2 × 2 matrix:

Ax Ay = AxBy − BxAy Bx By

For a 3 × 3 matrix:

Ax Ay Az Ax Ay +AxByCz + AyBzCx + AzBxCy Bx By Bz Bx By =

Cx Cy Cz Cx Cy −CxByAz − CyBzAx − CzBxAy

Using a co-factor expansion of the determinant:

Ax Ay Az By Bz Bx Bz Bx By Bx By Bz = Ax − Ay + Az Cy Cz Cx Cz Cx Cy Cx Cy Cz

= Ax( ByCz − C yBz) − Ay(BxC z − Bz Cx)

+ Az(BxCy − ByCx)= ǫijkAiBjCk c 2004-2017 G. Anderson Electricity & Magnetism – slide 20 / 68 Northeastern Illinois Vector Cross Product University Cartesian Basis Vectors:

î × ˆj = kˆ, ˆj × kˆ = î, kˆ × î = ˆj. (cyclic) C Determinant form: î ˆj kˆ B A × B = Ax Ay Az θ A B B B x y z

By components: | C | = | A || B | sin θ

(A × B)=(AyBz − AzBy,AzBx − AxBz,AxBy − AyBx)

(A × B)i = ǫijkAjBk c 2004-2017 G. Anderson Electricity & Magnetism – slide 21 / 68 Northeastern Illinois The Vector Cross Product University

Note that the cross product is anti-commutative

A × B = −B × A

From which follows the old McDonald formula:

ei × ei =0

Example Cross Products in Mechanics L = r × p N = r × F

c 2004-2017 G. Anderson Electricity & Magnetism – slide 22 / 68 Northeastern Illinois Levi-Civita Tensor University

The Levi-Civita tensor is completely antisymmetric:

ǫijk = −ǫikj = ǫkij

In general 1 ijk even permutation of 123 ǫijk =  −1 ijk odd permutation of 123  0 otherwise The cross product in terms of ǫ:

(A × B)i = ǫijkAjBk Example

(A × B)1 = ǫ1jkAjBk = ǫ11kA1Bk + ǫ12kA2Bk + ǫ13kA3Bk = 0+ ǫ123A2B3 + ǫ132A3B2 = A2B3 − A3B2 c 2004-2017 G. Anderson Electricity & Magnetism – slide 23 / 68 Northeastern Illinois Active Learning University

List the non-vanishing values of ǫijk.

c 2004-2017 G. Anderson Electricity & Magnetism – slide 24 / 68 Northeastern Illinois Active Learning University

List the non-vanishing values of ǫijk.

ǫ123 = 1 ǫ321 = −1 ǫ231 = 1 ǫ213 = −1 ǫ312 = 1 ǫ132 = −1

c 2004-2017 G. Anderson Electricity & Magnetism – slide 24 / 68 Northeastern Illinois Active Learning University

List the non-vanishing values of ǫijk.

ǫ123 = 1 ǫ321 = −1 ǫ231 = 1 ǫ213 = −1 ǫ312 = 1 ǫ132 = −1

Can you express this number as a factorial?

c 2004-2017 G. Anderson Electricity & Magnetism – slide 24 / 68 Northeastern Illinois Active Learning University

List the non-vanishing values of ǫijk.

ǫ123 = 1 ǫ321 = −1 ǫ231 = 1 ǫ213 = −1 ǫ312 = 1 ǫ132 = −1

Can you express this number as a factorial?

3 · 2 · 1=3!

c 2004-2017 G. Anderson Electricity & Magnetism – slide 24 / 68 Northeastern Illinois Scalar Triple Product University

A · (B × C)=(A × B) · C Proving identities is simple with ǫ:

A · (B × C) = Ai(B × C)i

= AiǫijkBjCk

= ǫkijAiBjCk

= (A × B)kCk = (A × B) · C

c 2004-2017 G. Anderson Electricity & Magnetism – slide 25 / 68 Northeastern Illinois The Kronecker Delta Tensor University

The Kronecker delta tensor is completely symmetric:

δij = δji

It vanishes unless both indices are the same: 1 i = j δ = ij 0 i =6 j

Written as a matrix:

δ11 δ12 δ13 1 0 0 δ =  δ21 δ22 δ23  =  0 1 0  δ δ δ 0 0 1  31 32 33   

Note that: Vi = δijVj,

Useful Identity: ǫijkǫklm = δilδjm − δimδjl

c 2004-2017 G. Anderson Electricity & Magnetism – slide 26 / 68 Northeastern Illinois Identity: Vector Triple Product University

Vector triple product:

A × (B × C)= B(A · C) − C(A · B)

Proof:

[A × (B × C)]i = ǫijkAj(B × C)k

= ǫijkAjǫklmBlCm

= ǫijkǫklmAjBlCm

= (δilδjm − δimδlj)AjBlCm

= AjBiCj − AjBjCi

= Bi(A · C) − Ci(A · B)

c 2004-2017 G. Anderson Electricity & Magnetism – slide 27 / 68 Northeastern Illinois University

Maxwell’s Equations

Vector Algebra Rotation of Vectors Rotation in 2D Rotation in 3D Eular Angles Tensors Parity Parity Continued Rotation of Vectors Parity

Grad & Div Curl & Stokes’ Thm. More Vector Calculus Curvilinear Coordinates

c 2004-2017 G. Anderson Electricity & Magnetism – slide 28 / 68 Northeastern Illinois Rotation of Vectors in 2D University

r = xî + yˆj = x′î′ + y′ˆj′ x′ = î′ · r = x(î′ · î)+ y(î′ · ˆj) = x cos φ + y cos(90◦ − φ) y y′ = x cos φ + y sin φ φ y′ = ˆj′ · r = x(ˆj′ · î)+ y(ˆj′ · ˆj) ′ φ x = x cos(90◦ + φ)+ y cos φ x = −x sin φ + y cos φ x′ cos φ sin φ x = y′ − sin φ cos φ y x′ x = R y′ y The two dimenstional rotation group, SO(2):

RT R = RRT = I, det R =1

c 2004-2017 G. Anderson Electricity & Magnetism – slide 29 / 68 Northeastern Illinois Rotations of Vectors in 3D University

Rotations about an arbitrary axis in 3D

′ x Rxx Rxy Rxz x ′ ′ x = Rx or  y  =  Ryx Ryy Ryz   y  z′ R R R z    zx zy zz    Any rotation matrix is an orthogonal matrix, RT = R−1:

RT R = RRT = I

Note that ′ ∂xi −1 −1 ∂xj Rij = = Rji , Rji = ′ = Rij ∂xj ∂xi

c 2004-2017 G. Anderson Electricity & Magnetism – slide 30 / 68 Northeastern Illinois Euler Angles and SO(3) University

Rotation around the z axis: cos φ sin φ 0 Rz(φ)=  − sin φ cos φ 0  0 01   Rotation around the x axis: 1 0 0 Rx(θ)=  0 cos θ sin θ  0 − sin θ cos θ   Three Euler Angles

R(ψ,θ,φ)= Rz′′ (ψ)Rx′ (θ)Rz(φ)

c 2004-2017 G. Anderson Electricity & Magnetism – slide 31 / 68 Northeastern Illinois Deﬁnitions: Scalars, Vectors, & Tensors University

Under a general rotation of the coordinate system: Scalar S′ = S Rank 1 tensor (Vector)

′ ′ Ai = RijAj, or A = RA Rank 2 tensor (Matrix )

′ ′ T Mij = RikRjlMkl, or M = RMR Rank 3 tensor ′ Tijk = RisRjtRkuTstu

c 2004-2017 G. Anderson Electricity & Magnetism – slide 32 / 68 Northeastern Illinois Parity aka Spatial Reﬂection University

Parity interchanges LH and RH coordinate systems.

(t, x) → (t, −x)

vectors which are odd (even) under parity are known as polar vectors, aka vectors (axial vectors, aka pseudovectors). x, p, J, E, and ∇ are vectors, while L, µ and B are axial-vectors. Classiﬁcation by parity transformations:

S → S Scalar P → −P Pseudoscalar E → −E Vectors B → B Pseudovectors

c 2004-2017 G. Anderson Electricity & Magnetism – slide 33 / 68 Northeastern Illinois Parity Continued University

Classiﬁcation by parity transformations:

S → S Scalar P → −P Pseudoscalar E → −E Vectors B → B Pseudovectors

• The inner product of two vectors, A · B is a • The scalar triple product A · (B × C) is a • The cross-product of two vectors, A × B is a • The gradient of a scalar ∇S is a

c 2004-2017 G. Anderson Electricity & Magnetism – slide 34 / 68 Northeastern Illinois Parity Continued University

Classiﬁcation by parity transformations:

S → S Scalar P → −P Pseudoscalar E → −E Vectors B → B Pseudovectors

• The inner product of two vectors, A · B is a scalar • The scalar triple product A · (B × C) is a pseudoscalar • The cross-product of two vectors, A × B is a pseudovector • The gradient of a scalar ∇S is a vector

c 2004-2017 G. Anderson Electricity & Magnetism – slide 34 / 68 Northeastern Illinois Parity University z

z x

x ←→ −x

c 2004-2017 G. Anderson Electricity & Magnetism – slide 35 / 68 Northeastern Illinois University

Maxwell’s Equations

Vector Algebra Rotation of Vectors

Grad & Div Gradient of Scalar Gradient Theorem Gradient, Divergence & Divergence Divergence & Flux Gauss’s Thm. Gauss’s Theorem Pf. Gauss’s Theorem Application to Fluid Dynamics Curl & Stokes’ Thm. More Vector Calculus Curvilinear Coordinates

c 2004-2017 G. Anderson Electricity & Magnetism – slide 36 / 68 Northeastern Illinois Gradient of a Scalar Function University

In three dimensions the gradient operator is deﬁned as: ∂ ∂ ∂ ∇ = ˆi + ˆj + kˆ Cartesian ∂x ∂y ∂z Coordinates

Acting on a scalar function f(x), ∇ produces a vector:

∂f ∂f ∂f ∇f = ˆi + ˆj + kˆ . gradient of f ∂x ∂y ∂z

∇f transforms like a vector (under rotation of Cartesian axes:)

3 ′ ∂f ∂xj ∂f (∇ f)i = ′ = ′ = Rij(∇f)j ∂x ∂x ∂xj i Xj=1 i

c 2004-2017 G. Anderson Electricity & Magnetism – slide 37 / 68 Northeastern Illinois The Gradient Theorem University

Consider a inﬁnitesimal displacement:

dx = ˆidx + ˆjdy + kˆdz

The change in φ(x) from x → x + dx:

∂φ ∂φ ∂φ dφ = φ(x + dx) − φ(x)= dx ∂x + dy ∂y + dz ∂z = dx ·∇φ

In any coordinate system:

dφ = dx ·∇φ.

For any curve C, with endpoints i and f:

φ(f) − φ(i)= ∇φ · dr ZC

c 2004-2017 G. Anderson Electricity & Magnetism – slide 38 / 68 Northeastern Illinois Divergence of a Vector Function University

′ The ∇ operator is a vector, ∇i = Rij∇j ∂ ∂ ∂ ∇ = ˆi + ˆj + kˆ ∂x ∂y ∂z

The divergence of a vector function F(x) is a scalar.

Fx(x) ∂ ∂ ∂ ∇· F(x) = ( ∂x , ∂y , ∂z )  Fy(x)  F (x)  z 

∂Fx(x) ∂Fy(x) ∂Fz(x) = ∂x + ∂y + ∂z .

This is invariant under rotations of the coordinate system.

c 2004-2017 G. Anderson Electricity & Magnetism – slide 39 / 68 Northeastern Illinois Physical Interpretation of Divergence University

Notation: {eˆ1, eˆ2, eˆ3} = {ˆi,ˆj, kˆ}

ǫ z + 2 3 1 F · dA = i=1 Fi(x + 2 ǫ eˆi) eˆ3 H P 1 2 − Fi(x − ǫ eˆi) ǫ eˆ2 2 eˆ1 3 ∂Fi 3 − ǫ = ǫ x 2 i=1 ∂xi P 3 ǫ ǫ = (∇· F)ǫ z − 2 x + 2 y − ǫ/2 y + ǫ/2

Divergence is the net ﬂux out of a volume, per unit volume, in the limit of inﬁntesimal volume. 1 ∇· F = lim F · dA V →0 V IS

c 2004-2017 G. Anderson Electricity & Magnetism – slide 40 / 68 Northeastern Illinois Towards Gauss’s Theorem University

1 ∇· F = lim F · dA V →0 V IS For an inﬁntesimal volume dV , bounded by a surface dS:

∇· F dV = F · dA IdS Sum LHS and RHS over connected inﬁntesimal volumes:

∇· Fd3x = F · dA ZV IS

c 2004-2017 G. Anderson Electricity & Magnetism – slide 41 / 68 Northeastern Illinois Gauss’s Theorem (Divergence Theorem) University

Outward ﬂux through a closed surface equals integral of divergence in the enclosed volume.

F · dA = ∇· Fd3x IS ZV

Gauss’s theorem, aka. Green’s theorem, divergence theorem

c 2004-2017 G. Anderson Electricity & Magnetism – slide 42 / 68 Northeastern Illinois Application to Fluid Dynamics University

For a fluid with density ρ, pressure p and velocity v, conservation of mass can be written: ∂ρ + ∇· (ρv)=0 ∂t Using Gauss’ theorem d ρv · dA = − ρd3x Is dt ZV The vorticity of a fluid is defined as: ∇× v. Around a whirlpool:

v · dℓ =06 IC

c 2004-2017 G. Anderson Electricity & Magnetism – slide 43 / 68 Northeastern Illinois University

Maxwell’s Equations

Vector Algebra Rotation of Vectors

Grad & Div Curl & Stokes’ Thm. Curl Curl & Circulation Curl & Stokes’ Theorem Stokes’ Thm. Pf. Stokes’ Theorem The Helmholtz Theorem More Vector Calculus Curvilinear Coordinates

c 2004-2017 G. Anderson Electricity & Magnetism – slide 44 / 68 Northeastern Illinois Curl of a Vector Function University

The curl of F(x), ∇× F is a vector.

ˆi ˆj kˆ ∂ ∂ ∂ ∂Fk ∇× F = (∇× F)i = ǫijk ∂x ∂y ∂z ∂x F F F j x y z

The curl is a measure of vorticity or circulation. It tells you by how much a vector ﬁeld curls around a point.

c 2004-2017 G. Anderson Electricity & Magnetism – slide 45 / 68 Northeastern Illinois Physical Interpretation of Curl University

j ǫ

ǫ dP F · dl = ǫFi(x − 2 eˆj) ǫ ǫ xj + 2 H + ǫFj(x + 2 eˆi) ǫ − ǫFi(x + eˆj) xj 2 ǫ ǫ − ǫFj(x − 2 eˆi)

ǫ b xj − 2 − ǫ ǫ ∂Fj ∂F xi 2 xi + 2 = − i ǫ2 ∂xi ∂xj 2 i = (∇× F )kǫ xi

With the normal vector nˆ = kˆ: 1 nˆ · (∇× F)= lim F · dl A→0 A IC

c 2004-2017 G. Anderson Electricity & Magnetism – slide 46 / 68 Northeastern Illinois Towards Stokes’ Theorem University

1 nˆ · (∇× F)= lim F · dl A→0 A IC For an inﬁnitesimal area dA bounded by dC

dA · (∇× F)= F · dl IdC Sewing inﬁntesimal areas together: joining them at the boundaries:

(∇× F) · dA = F · dl ZS IC

c 2004-2017 G. Anderson Electricity & Magnetism – slide 47 / 68 Northeastern Illinois Kelvin-Stokes Theorem University

Circulation of a vector function around a closed loop is equal to the ﬂux of the curl through the surface bounded by C.

F · dl = (∇× F) · dA IC ZS

The line integral of a vector around a closed curve is equal to the integral of the normal component of its curl over a surface bounded by the closed curve.

c 2004-2017 G. Anderson Electricity & Magnetism – slide 48 / 68 Northeastern Illinois The Helmholtz Theorem University

Sufficiently bounded functions are uniquely determined by their curl and divergence. If H(x) is differentiable at all points and space, define: d(x)= ∇· H and c(x)= ∇× H If d and c approach zero faster than r−2 as r →∞ and if H approaches zero in the same limit, then:

H(x)= −∇ψ + ∇× A

where d(x′) ψ(x)= d3x′ Z 4π | x − x′ | c(x′) A(x)= d3x′ Z 4π | x − x′ |

c 2004-2017 G. Anderson Electricity & Magnetism – slide 49 / 68 Northeastern Illinois University

Maxwell’s Equations

Vector Algebra Rotation of Vectors

Grad & Div Curl & Stokes’ Thm. More Vector Calculus More Vector Calculus ∇ & Chain Rule Vanishing Products Other ∇ Identities The Laplacian Curvilinear Coordinates

c 2004-2017 G. Anderson Electricity & Magnetism – slide 50 / 68 Northeastern Illinois ∇ and the Chain Rule University

The Leibniz rule applies for diﬀerentiating products:

ˆ ∂ ˆ ∂ ˆ ∂ ∇(fg) = i∂x(fg)+ j∂y (fg)+ k∂z (fg)

ˆ∂g ˆ∂g ˆ ∂g ˆ∂f ˆ∂f ˆ ∂f = f i∂x + j∂y + k∂z + g i ∂x + j ∂y + k ∂z = f∇g + g∇f

c 2004-2017 G. Anderson Electricity & Magnetism – slide 51 / 68 Northeastern Illinois Vanishing Products University

The curl of a gradient vanishes: ∂2f (∇×∇f)i = ǫijk =0 ∂xj∂xk The divergence of a curl vanishes 2 ∂ Fk ∇· (∇× F)= ǫijk =0 ∂xi∂xj

c 2004-2017 G. Anderson Electricity & Magnetism – slide 52 / 68 Northeastern Illinois Other ∇ Identities University

Derivatives of Products ∇(fg) = f∇g + g∇f ∇· (gF) = g(∇· F)+(∇g) · F ∇× (gF) = g(∇× F)+(∇g) × F ∇· (F × G) = (∇× F) · G − F · (∇× G) Products of Derivatives ∇× (∇f) =0 ∇· (∇× F) =0 ∇× (∇× F) = ∇(∇· F)+ ∇2F

c 2004-2017 G. Anderson Electricity & Magnetism – slide 53 / 68 Northeastern Illinois The Laplacian University

Recall the gradient of a scalar f: ∂f ∂f ∂f ∇f = ˆi + ˆj + kˆ ∂x ∂y ∂z The divergence of a gradient: ∂2f ∂2f ∂2f ∇2f ≡∇·∇f = + + ∂x2 ∂y2 ∂z2 The Laplacian operator in Cartesian coordinates: ∂2 ∂2 ∂2 ∇2 = + + ∂x2 ∂y2 ∂z2

c 2004-2017 G. Anderson Electricity & Magnetism – slide 54 / 68 Northeastern Illinois University

Maxwell’s Equations

Vector Algebra Rotation of Vectors

Grad & Div Curl & Stokes’ Thm. More Vector Calculus Curvilinear Coordinates Curvilinear Coordinates Curvilinear Coordinates for Rn Curvilinear Coordinates for R3 Jacobian Plane Polar Coordinates in 2 (2D) R Cylindrical Coord. Spherical Coord. Gradient Gradient II Divergence Curl Curl c 2004-2017 In General G. Anderson Electricity & Magnetism – slide 55 / 68 Coordinates II Northeastern n Illinois Curvilinear Coordinates for R University

Curvilinear coordinates: curved line coordinate systems for ﬂat (Euclidean) space. Curvilinear coordinates can be converted (transformed) to the standard Cartesian coordinate system.

u1 = u1(x,y,z,...) u2 = u2(x,y,z,...) u3 = u3(x,y,z,...) ...... un = un(x,y,z,...) curvilinear coordinates, coined by the French mathematician Gabriel Lam´ewho developed a general theory of curvilinear coordinates. c 2004-2017 G. Anderson Electricity & Magnetism – slide 56 / 68 Northeastern 3 Illinois Curvilinear Coordinates for R University

Generalized Coordinates: (u1,u2,u3)

Cartesian: (x,y,z) ds Spherical: (r,θ,φ) 3

h Cylindrical: (r,φ,z)

3 du

3 eˆ3 Volume element:

eˆ2 1 e du ˆ1 1 h dV =(h1du1)(h2du2)(h3du3) = 1 ds du2 ds2 = h2 length element: dsi, scale factor: hi.

Inﬁntesimal displacement:

ds = eˆ1h1du1 + eˆ2h2du2 + eˆ3h3du3

c 2004-2017 G. Anderson Electricity & Magnetism – slide 57 / 68 Northeastern Illinois Jacobian Matrix and Determinant University

Given u = f(x), the Jacobian matrix is deﬁned by:

∂u1 ∂u1 ∂x1 ··· ∂xn . . . J(x1,...,xn)=  . .. .  ∂un ··· ∂un  ∂x1 ∂xn  The determinant of the Jacobian matrix, or Jacobian determinant: ∂(u ,...,u ) du ...du = 1 n dx ...dx 1 n ∂(x ,...,x ) 1 n 1 n

gives ratio of n-dimensional, inﬁntesimal volumes.

c 2004-2017 G. Anderson Electricity & Magnetism – slide 58 / 68 Northeastern 2 Illinois Plane Polar Coordinates in (2D) R University

Cartesian coordinates Plane polar coordinates

y (x + dx, y + dy) y (r + dr, φ + dφ)

x = r cos φ y = r sin φ (x, y) (r, φ) φ x x

x = xˆi + yˆj x = rˆr ˆ ˆ ds = dxi + dyj ds = dr eˆr + rdφ eˆφ dA = dxdy dA = rdrdφ

c 2004-2017 G. Anderson Electricity & Magnetism – slide 59 / 68 Northeastern Illinois Cylindrical Coordinates (r,φ,z) University

z kˆ z x = r cos φ r φˆ RH coordinates: (eˆi × eˆj = ǫijkeˆk) y = r sin φ z = z P ˆr u1 = r, u2 = φ, u3 = z

y Scale factors: r φˆ φ hr =1, hφ = r, hz =1 x

Position: x = rˆr + zkˆ Displacement: ds = ˆrdr + φˆrdφ + kˆdz Volume element: dV = rdrdφdz

c 2004-2017 G. Anderson Electricity & Magnetism – slide 60 / 68 Northeastern Illinois Spherical Coordinates (r,θ,φ) University

z ˆr x = r sin θ cos φ φˆ RH coordinates: (eˆi × eˆj = ǫijkeˆk) y = r sin θ sin φ P z = r cos θ θ θˆ u1 = r, u2 = θ, u3 = φ r y Scale factors: φˆ φ hr =1, hθ = r, hφ = r sin θ x Position: x = rˆr Displacement: ds = ˆrdr + θˆrdθ + φˆr sin θdφ Volume element: dV = r2 dr sin θ dθdφ = r2drdΩ

c 2004-2017 G. Anderson Electricity & Magnetism – slide 61 / 68 Northeastern Illinois Gradient in General Coordinates University

General Coordinates: (u1,u42,u3) Inﬁntesimal Displacement:

ds = eˆ1h1du1 + eˆ2h2du2 + eˆ3h3du3

Gradient deﬁned by: df = ∇f · ds, or

∇f · ds = (∇f)1h1du1 + (∇f)2h2du2 + (∇f)3h3du3

Also, by direct diﬀerentiation: ∂f ∂f ∂f df = du1 + du2 + du3 ∂u1 ∂u2 ∂u3 1 ∂f (∇f)i = hi ∂ui

c 2004-2017 G. Anderson Electricity & Magnetism – slide 62 / 68 Northeastern Illinois Gradient in Curvilinear Coordinates University

Inﬁntesimal displacement:

ds = eˆ1h1du1 + eˆ2h2du2 + eˆ3h3du3 1 ∂f (∇f)i = hi ∂ui

Cylindrical: hr = 1, hφ = r, hz = 1:

∂f 1 ∂f ∂f ∇f = ˆr + φˆ + kˆ ∂r r ∂φ ∂z

Spherical: hr = 1, hθ = r, hφ = r sin θ:

∂f 1 ∂f 1 ∂f ∇f = ˆr + θˆ + φˆ ∂r r ∂θ r sin θ ∂φ

c 2004-2017 G. Anderson Electricity & Magnetism – slide 63 / 68 Northeastern Illinois Divergence in General Coordinates University

3 du dV = h1h2h3du1du2du3 3

e ˆ3 ∇· FdV = ﬂux out of faces eˆ2 eˆ1 1 du P 1 = F · dA h S h2 du2 H

∇· FdV = [F1h2h3 |u1+du1 −F1h2h3 |u1 ] du2du3

+ [F2h1h3 |u2+du2 −F2h1h3 |u2 ] du1du3

+ [F3h1h2 |u3+du3 −F3h1h2 |u3 ] du1du2

1 ∂(F1h2h3) ∂(F2h1h3) ∂(F3h1h2) ∇· F = h h h ∂u + ∂u + ∂u 1 2 3 h 1 2 3 i

c 2004-2017 G. Anderson Electricity & Magnetism – slide 64 / 68 Northeastern Illinois Curl in General Coordinates University

(u1 + du1, u2 + du2) 1 nˆ · (∇× F)= lim F · dℓ A→0 A IC eˆ2

2 1 (∇× F)3 = du (h1du1)(h2du2) 2 eˆ1 h h = + (F2h2 |u1+du1 −F2h2 |u1 ) du2 2

(u1, u2) ds ds1 = h1 du − (F1h1 |u2+du2 −F1h1 |u2 ) du1 1 i Diﬀerence equation: X |u+du −X |u=(∂X/∂u)du

1 ∂ ∂ (∇× F)3 = (F2h2) − (F1h1) h1h2 ∂u1 ∂u2

c 2004-2017 G. Anderson Electricity & Magnetism – slide 65 / 68 Northeastern Illinois Curl In General Coordinates II University

In General:

eˆ1/h2h3 eˆ2/h3h1 eˆ3/h1h2 ∇× F = ∂/∂u ∂/∂u ∂/∂u 1 2 3 h F h F h F 1 1 2 2 3 3

c 2004-2017 G. Anderson Electricity & Magnetism – slide 66 / 68 Northeastern Illinois Laplacian in General Coordinates University

Gradient of a scalar: 1 ∂f 1 ∂ (∇f)i = ⇒ ∇f = eˆi hi ∂ui hi ∂ui Divergence of a vector

1 ∂(F1h2h3) ∂(F2h1h3) ∂(F3h1h2) ∇· F = h h h ∂u + ∂u + ∂u 1 2 3 h 1 2 3 i = 1 ∂(Fihj hk) h1h2h3 cyclic(ijk) ∂ui P Divergence of the gradient, ∇·∇f:

1 ∂ h h ∂f ∇2f = j k h1h2h3 ∂ui hi ∂ui cyclic(Xijk)

c 2004-2017 G. Anderson Electricity & Magnetism – slide 67 / 68 Northeastern Illinois Next University

Lecture 04: Electrostatics I

c 2004-2017 G. Anderson Electricity & Magnetism – slide 68 / 68