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Lecture 2: Review of Vector CalculusInstructor: Dr. Gleb V. Tcheslavski Contact: Office Hours: Room 2030Class web site: Vector normProperties:Example: v = (1, 2, 3)NameSymbolvalueL1 – norm|v|16L2 – norm|v|2141/2 ? 3.74L3 – norm|v|362/3? 3.3L4 – norm|v|421/471/2? 3.15L? – norm|v|?3(2.2.1)(2.2.2)(2.2.3)(2.2.4)(2.2.5)(2.2.6)Norm(x,p) Vector sum Scalar (dot) product(2.4.1)(2.4.2)(2.4.3)Definitions:Scalar projection:(2.4.5)(2.4.6)dot(A,B)Property:(2.4.4) Vector (cross) productcross(A,B)(2.5.1)Definitions:Properties:In the Cartesian coordinate system:(2.5.2)(2.5.4)(2.5.5)(2.5.3) Triple products1. Scalar triple product:2. Vector triple product:(2.6.1)(2.6.2)Note: (2.6.1) represents a circular permutation of vectors. Q: A result of a dot product is a scalar, a result of a vector product is a vector.What is about triple products? Vector fieldsA vector field is a map f that assigns each vector x a vector function f(x).From Wolfram MathWorldA vector field is a construction, which associates a vector to every point in a (locally) Euclidean space.A vector field is uniquely specified by giving its divergence and curl within a region and its normal component over the boundary. Coordinate systems In a 3D space, a coordinate system can be specified by the intersection of 3 surfaces. An orthogonal coordinate system is defined when these three surfaces are mutually orthogonal at a point.A general orthogonal coordinate system: the unit vectors are mutually orthogonalThe cross-product of two unit vectors defines a unit surface, whose unit vector is the third unit vector. Most commonly used coordinate systems– Cartesian; (b) – Cylindrical; (c) – Spherical.In Cartesian CS, directions of unit vectors are independent of their positions;In Cylindrical and Spherical systems, directions of unit vectors depend on positions. Coordinate systems: CartesianAn intersection of 3 planes:x = const; y = const; z = constProperties:(2.10.1)(2.10.2)(2.10.3)An arbitrary vector:(2.