relation_to_phy

relation_to_phy - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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V15. Relation to Physics The three theorems we have studied: the divergence theorem and Stokes' theorem in space, and Green's theorem in the plane (which is really just a special case of Stokes' theo- rem) are widely used in physics and continuum mechanics, in the study of fields, potentials, heat flow, wave motion in liquids, gases, and solids, and thermodynamics, to name some of the uses. Often partial differential equations which model some physical situation are de- rived using the vector integral calculus theorems. This section is devoted to a brief account of where you will first meet the theorems: in electromagnetic theory. 1. Symbolic notation: the del operator To have a compact notation, wide use is made of the symbolic operator "del" (some call it "nabla") : a aM Recall that the "product" of - ax and the function M(x, y, z) is understood to be -. Then ax we have af af af grad f = Vf = -i+-j ax +-k ay az The divergence is sort of a symbolic scalar product: if F = M i + Nj + Pk , while the curl, as we have noted, as a symbolic cross-product: i j k a a curlF = VxF= . MNP Notice how this notation reminds you that V . F is a scalar function, while V x F is a vector function. We may also speak of the Laplace operator (also called the "Laplacian"), defined by Thus, Laplace's equation may be written: v2 f = 0. (This is for example the equation satisfied by the potential function for an electrostatic field, in any region of space where there are no charges; or for a gravitational field, in a region of space where there are no masses.) In this notation, the divergence theorem and Stokes' theorem are respectively
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2 V. VECTOR INTEGRAL CALCLUS Two important relations involving the symbolic operator are: (7) curl (grad f) = 0 div curl F = 0 (7') VxVf = 0 V-VxF = 0 The first we have proved (it was part of the criterion for gradient fields); the second is an easy exercise. Note however how the symbolic notation suggests the answer, since we know that for any vector A, we have AxA = 0, A.AxF = 0, and (7') says this is true for the symbolic vector V as well.
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relation_to_phy - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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