Lecture 13 Basic Laws of Vector Algebra Scalars: e.g. 2 gallons, $1,000, 35ºC Vectors: e.g. velocity: 35mph heading south 3N force toward center

Lecture 13 Cartesian coordinate system x y z A  

Lecture 13 Vector addition and subtraction C = B+A = A +B, |C| = |B+A| = |A| +|B|, A B C A B C head-to-tail rule parallelogram rule A B D D = A - B = -(B –A), A B D’ = (B-A)

Lecture 13 position and distance x y z A B D = A - B = -(B –A),

Lecture 13 Vector multiplication 1. simple product 2. scalar product (dot product)

Lecture 13 Properties of scalar product (dot product) a) commutative property b) Distributitve property

Lecture vector product (cross product) a) anticommutative property b) Distributitve property c)

Lecture vector product (cross product)

Lecture 13 Example vectors and angles In Cartesian coordinate, vector A is directed from origin to point P1(2,3,3), and vector B is directed from P1 to pint P2(1,-2,2). Find: (a) Vector A, its magnitude |A|, and unit vector a (b) the angle that A makes with the y-axis (c) Vector B (d) the angle between A and B (e) perpendicular distance from origin to vector B

Lecture Scalar and vector triple product a) scalar triple product b) vector triple product

Lecture 13 Example vector triple product

Lecture 14 Cartesian coordinate system x y z dl  

Lecture 14 Cartesian coordinate system x y z directions of area

Lecture 14 Cylindrical coordinate system z x y

Lecture 14 the differential areas and volume z x y

Lecture 14 Example: cylindrical area z x y

Lecture 14 Spherical coordinate system

Lecture 14 differential volume in Spherical coordinate system

Lecture 14 Examples (1) Find the area of the strip (2) A sphere of radius 2cm contains a volume charge density Find the total charge contained in the sphere

Lecture 15 Cartesian to cylindrical transformation

Lecture 15 Cartesian to cylindrical transformation

Lecture 15 Cartesian to cylindrical transformation

Lecture 15 Cartesian to Spherical transformation

Lecture 15 Cartesian to Spherical transformation

Lecture 15 Cartesian to Spherical transformation

Lecture 15 Distance between two points:

Lecture 16 Gradient in Cartesian Coordinates Gradient: differential change of a scalar The direction ofis along the maximum increase of T.

Lecture 16 Example of Gradient in Cartesian Coordinates Find the directional derivative ofalong the direction and evaluate it at (1, -1,2).

Lecture 16 Gradient operator in cylindrical Coordinates

Lecture 16 Gradient operator in cylindrical Coordinates z x y

Lecture 16 Gradient operator in Spherical Coordinates

Lecture 16 Properties of the Gradient operator

Lecture 17 Flux in Cartesian Coordinates

Lecture 17 Flux in Cartesian Coordinates

Lecture 17 Definition of divergence in Cartesian Coordinates

Lecture 17 Properties of divergence IfNo net flux on any closed surface. Divergence theorem

Lecture 17 Divergence in Cylindrical Coordinates z x y

Lecture 17 Divergence in Cylindrical Coordinates z x y

Lecture 17 Divergence in Spherical Coordinates

Lecture 17 Divergence in Spherical Coordinates

Lecture 17 Divergence in Spherical Coordinates

Lecture 18 Circulation of a Vector

Lecture 18 Circulation of a Vector

Lecture 18 Curl in Cartesian Coordinates

Lecture 18 Vector identities involving the curl Stokes’s theorem

Lecture 18 Curls in Rectangular, Cylindrical and Spherical Coordinates

Lecture 18 Laplacian Operator of a scalar

Lecture 18 Laplacian Operator of a vector