Chapter 1 Vector analysis August 22 Elementary approach 1. 1 Definitions, elementary approach Scalar quantities: Quantities having magnitude only. Length, mass, time, temperature, energy. Vector quantities: Quantities having both magnitude and directions. Displacement, velocity, acceleration, momentum, angular momentum, electric field, magnetic field, dipoles. Tensor quantities: (Tensors of rank n) moment of inertia, electric permittivity, nonlinear susceptibility . Geometrical representation of a vector: An arrow. Addition and subtraction: Vector addition is commutative and associative.

Algebraic representation of a vector: component, projection direction cosine unit vector, basis more preferred form most convenient for doing algebra magnitude (length, norm)

1. 2 Rotation of the coordinate axes V scalar product (more later) Kronecker delta symbol Einstein’s summation convention: Any repeated indices are summed over, unless otherwise specified. Greatly simplifies many expressions. Rotation matrix aij represents the transformation between the two sets of coordinates. Its elements are the projections between the two sets of bases. Orthogonality conditions: aij Orthonormal transformation

Components of a vector after transformation: Question: Is electric current (I) a vector? Redefinition of a vector quantity: A quantity is a vector if its components transform as under an orthonormal transformation specified by aij. For tensors: Generalization of vectors. Vectors can also be Complex quantities. Functions. Multi-dimensions or infinite dimensions.

Read: Chapter 1: 1-2 Homework: 1.1.1, 1.2.1(a) Due: September 2

August 24,26 Scalar and vector products 1. 3 Scalar or dot product B A q Geometric definition: Scalar product is commutative and distributive. Algebraic definition: ( , orthonormal) Invariance of the scalar product under an orthogonal transformation: We thus proved that A·B is indeed a scalar.

1. 4 Vector or cross product B A q C Geometric definition: Meaning: area of the parallelogram formed by A and B. Cross product is anticommutative: Algebraic definition: Levi-Civita symbol (antisymmetric tensor) eijk: 1 2 3 + - Look at your watch.

Cross product is based on the nature of our 3-dimentional space. Cross product in determinant form: Cross product is based on the nature of our 3-dimentional space. Matrix representation of cross product: Example:

= the volume enclosed by the parallelepiped defined by A, B and C. 1. 5 Triple scalar product, triple vector product Triple scalar product : = the volume enclosed by the parallelepiped defined by A, B and C.

Can be proved by brute force (though not trivial). Triple cross product : Can be proved by brute force (though not trivial). C B q B×C A The “bac-cab” rule.

Read: Chapter 1: 3-5 Homework: 1.3.3, 1.4.7,1.5.12 Due: September 2

Reading: Proof that C=A×B is a vector:

About matrices: Inverse matrix A-1 : Transpose matrix Ã: For orthonormal matrices: About determinates: Minor Mij: removing the ith row and the jth column Cofactor Cij : (-1)i+j Mij Expansion of a determinant: Calculation of A-1: For orthonormal matrices with |A|=1:

August 29, 31 Gradient, divergence and curl 1. 6 Gradient,  Definition: In a 3D Cartesian coordinate system, Example: Central force: Proof that j is a vector:

Geometrical interpretations: 1) The function j has the steepest change along the direction of its gradient. 2) j is perpendicular to the surface of

1. 7 Divergence ,· Definition: In a 3D Cartesian coordinate system, Examples: Physical meaning: is the net outflow flux of j per unit volume. B is solenoidal if

1. 8 Curl,  Definition: In a 3D Cartesian coordinate system, Example: Physical meaning: Set the coordinate system so that z is along at an arbitrarily chosen point. Suppose the coordinates of that point is then (x0, y0, z0 ). is the total circulation of V per unit area. V is irrotational if

1. 9 Successive applications of  Example: 1) 1. 9 Successive applications of  2) 3) 4)

More about Laplacian:

Reading: Physical meaning of Laplacian: Laplacian measures the difference between the average value of the field around a point and the value of the field at that point. If then j cannot increase or decrease in all directions.

Read: Chapter 1: 6-9 Homework: 1.6.3,1.6.4,1.7.5,1.8.4,1.8.11,1.8.13,1.8.14,1.9.7 Due: September 9

September 2,7 Gauss’ theorem and Stokes’ theorem 1. 10 Vector integration A B Line integral: Circulation: (mostly used line integral) If , the line integral is independent of the path between A and B: The circulation of F around any loop is then 0: Surface integral: (mostly used surface integral, flux) Volume integral:

Integral definition of gradient, divergence and curl: z dy dz y x dx Proof: let

1. 11 Gauss’ theorem Gauss’ theorem : (Over a simply connected region.) Proof: For a differential cube, Sum over all differential cubes, at all interior surfaces will cancel, only the contributions from the exterior surfaces remain.

then use Gauss’ theorem. Green’ theorem : Proof: then use Gauss’ theorem. Variant: Alternate forms of Gauss’ theorem : These can also be obtained from the integral definition of gradient, divergence and curl. Example of Gauss’ theorem: Gauss’ law of electric field

1. 12 Stokes’ theorem Stokes’ theorem : (Over a simply connected region. The surface does not need to be flat.) Proof: Set the coordinate system so that x is along ds at an arbitrarily chosen point on the surface. Suppose the coordinates of that point is then (x0, y0, z0 ). Sum over all differential squares, at all interior lines will cancel, only the contributions from the exterior lines remain.

Alternate forms of Stokes’ theorem : Example of Stokes’ theorem: Faraday’s induction law Stokes’ theorem for a curve on the x-y plane: Green’s theorem

Read: Chapter 1: 10-12 Homework: 1.10.5,1.11.2,1.11.6,1.12.3,1.12.10 Due: September 9

September 9 Gauss’ law and Dirac delta function 1. 14 Gauss’s law Electric field of a point charge: q S S′ Gauss’ law: Proof: 1) If S does not include q, using Gauss’ theorem,

2) If S includes q, construct a small sphere S' with radius d around the charge and a small hole connecting S and S’. q S S′ If S includes multiple charges, For a charge distribution, Using Gauss’ theorem,

1. 15 Dirac delta function Dirac delta function is defined such that Especially, Example: Using the Dirac delta function,

Dirac delta function (distribution) is the limit of a sequence of functions, such as The sequence of integrals has a limit: We write it as implying that we are always doing the limit.

Properties of d (x): Proof:

Read: Chapter 1: 13-15 Homework: 1.13.8,1.13.9,1.14.3,1.15.7 Due: September 16

September 12 Helmholtz’s theorem 1. 16 Helmholtz’s theorem The uniqueness theorem : A vector is uniquely specified within a simply connected region by given its 1) divergence , 2) curl, and 3) normal component on the boundary. Proof:

Corollary: The solution of Laplacian equation is unique at a given boundary condition.

Helmholtz’s theorem (the fundamental theorem of vector calculus): Any rapidly decaying vector field (faster than 1/r at infinity) can be resolved into the sum of an irrotational vector field and a solenoidal vector field. Proof: Let us prove that any vector V can be decomposed as The explicit expressions of j and A are given later.

Read: Chapter 1: 16 Homework: 1.16.1 Due: September 23