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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)i + v2(t)j + v3(t)k  the derivative of v(t) is v’ = [v1’, v2’, v3’] = v1’i + v2’j + v3’k
(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 :

In case you're interested in learning more about Vectors, here's the full set of tutorials we have :

Vectors 1a ( Theory and Definitions: Introduction to Vectors; Vector, Scalar and Triple Products)
Introducing a vector, position vectors, direction cosines, different types of vectors, addition and subtraction of vectors. Vector and Scalar products. Scalar Triple product and Vector triple product and their properties. Components and projections of vectors.

Vectors 1b ( Solved Problem Sets: Introduction to Vectors; Vector, Scalar and Triple Products )
Solved examples and problem sets based on the above concepts. 

Vectors 2a ( Theory and Definitions: Vectors and Geometry ) Vectors and geometry. Parametric vectorial equations of lines and planes. Angles between lines and planes. Co-planar and collinear points. Cartesian equations for lines and planes in 3D. 

Vectors 2b ( Solved Problem Sets: Vectors and Geometry )

Solved examples and problem sets based on the above concepts.

Vectors 3a ( Theory and Definitions: Vector Differential and Integral Calculus ) Vector Differential Calculus. Derivative, curves, tangential vectors, vector functions, gradient, directional derivative, divergence and curl of a vector function; important formulas related to div, curl and grad. Vector Integral Calculus. Line integral, independence of path, Green's theorem, divergence theorem of Gauss, green's formulas, Stoke's theorems.

Vectors 3b ( Solved Problem Sets: Vector Differential and Integral Calculus ) - Solved examples and problem sets based on the above concepts.




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