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Vector Differential And Integral Calculus: Theory and Definitions - Differentiation of Vectors, Introduction to Div, Curl, Grad; Vector Integral Calculus; Green’s theorem in the plane; Divergence theorem of Gauss, etc.

                                                                               Vector Differential And Integral Calculus

Here's a quick outline of the topics we'll introduce in this tutorial :

Differentiation of Vectors

• If v (t) = [v1(t), v2(t), v3(t)] = v1(t)ijk  tijk
(u • v)u’ • v u • v’
(u x v)’ = u’ x v + u x v’.

Introduction to Div, Curl, Grad 

Divergence, Curl and Gradient

Vector Integral Calculus

The analogue of the definite integral of calculus is the line integral

Independence of path of a line integral in a domain D means that the integral of a given function over any path with endpoints A and B has the same value for all paths from A to B that lie in D; here A and B are fixed. An integral (1) is independent of path in D if and only if the differential form with continuous F1, F2, F3 is exact in D. Also, if curl F = 0, where F = [F1, F2, F3], has continuous first partial derivatives in a simply connected domain D, then the integral (1) is independent of path in D.

Integral Theorems

These are theorems which will be introduced. 

o Green’s theorem in the plane
o Divergence theorem of Gauss
o Green’s formulas
o Green’s formulas

Complete Tutorial :