Scalar fields and vector functions
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Vector calculus - Contour curves
Suppose we have a two-dimensional scalar field phi(x,y). Then a curve will be definted by each specific value that phi(x,y) can take. [Equation goes here - download the original to see it.] The curve is called a contour curve of the scalar field. [Equation goes here - download the original to see it.] (Note that the contour curves of a temperature field are called isotherms and the contour curves of a pressure field are called isobars.) Example [Equation goes here - download the original to see it.] Sketch the contours given by [Equation goes here - download the original to see it.] Solution [Equation goes here - download the original to see it.] This is the equation of the circle with centre the origin and radius 1. Similarly, [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] A contour map is a map of a surface given by its contours. If we make a contour map by taking a constant interval between each contour; e.g. [Equation goes here - download the original to see it.] where the difference between successive contours is always 10 units, then the rate of increase or decrease of a scalar field is related to the closeness of the contour curves. The closer the contour curves are together the faster the scalar field is changing. The idea of a contour curve can be generalised to 3-dimensions [Equation goes here - download the original to see it.] This will give pictorially a series of contour surfaces. For example, if Equation goes here - download the original to see it.] then the contour surfaces given by [Equation goes here - download the original to see it. for different values of k are a series of nested spheres. A contour surface can be defined for an n-dimensional scalar field [Equation goes here - download the original to see it.]
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Contents of
Scalar fields and vector functions
1 Vector calculus - Scalar Field
2 Vector calculus - Contour curves
3 Vector calculus - Vector Field
4 Vector calculus - Vector field lines
5 Differentiation of scalar and vector products.
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