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On completion of this module, students should be able to: a calculate vector and scalar derivatives of vector and scalar fields using the grad, div and curl operators in Cartesian and in cylindrical and spherical polar coordinates; b use suffix notation to manipulate Cartesian vectors and their derivatives; c calculate multiple integrals in two and three dimensions including changing variables using Jacobians; d calculate line and surface integrals and use the various integral theorems.

Vector calculus

Back to Discovery Themes. Log on. Evans leeds. This Module is approved as a Discovery Module. Discovery module overview.

Vector Fields

Module Summary Vector calculus is the extension of ordinary one-dimensional differential and integral calculus to higher dimensions. Vector Calculus: grad, div, curl and the operator. The directional derivative and Laplacian operators.


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Suffix notation: representation of vectors and their products using suffix notation. The Kronecker delta and alternating tensors. Grad, div and curl in suffix notation.

Vector Calculus - Application Center

Use of suffix notation to manipulate products and combinations of vector differentials. Double and triple integrals of scalars. We have fundamental theorems for one-dimensional curves , two-dimensional planar regions and surfaces , and three-dimensional volumes objects.

If you wanted to understand why these theorems are all so similar, you could do some research into how all four theorems are special cases of something called the generalized Stokes' theorem. Recognizing the similarity of the four fundamental theorems can help you understand and remember them.

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Math Insight

Here we summarize the theorems and outline their relationships to the various integrals you learned in multivariable calculus. In words, this means the line integral of the gradient of some function is just the difference of the function evaluated at the endpoints of the curve. Green's theorem relates a double integral over a region to a line integral over the boundary of the region.

Calculus 3 Lecture 15.1: INTRODUCTION to Vector Fields (and what makes them Conservative)

Outer boundaries must be counterclockwise and inner boundaries must be clockwise. Stokes' theorem relates a line integral over a closed curve to a surface integral.


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To check for proper orientation, use the right hand rule. The divergence theorem relates a surface integral to a triple integral. The integrand of the triple integral can be thought of as the expansion of some fluid.