ASE 366K - Vector Calculus Primer

ASE 366K - Vector Calculus Primer - 4 20 A ppendix. Vectors...

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420 Appendix. Vectors and Vector Calculus In a coordinate system with orthonormal vectors (i, j, k), we can find the coordinates for a x b in terms of the coordinates of a and the coordinates of b. ixj : k jxk : i kxi j Note: the vectors i, j, and k are oriented in such a way that they form the basis of a right-handed coordinate system. Using the above formulae, and the anti-commutative law, one obtains the cartesian coordinate expression for the cross product: a x b (a,i -f aoj -t a"k) x (b"i -F bai + b"k) (oob" - boa")i I (a"b* b"a,)i I (a*b, b,ao)k. Briefly, & X b (oob, boar,a"b* b"a*,a*bs b"ay) . An important formula which we shall use several times, and which can be verified by direct (and repeated) use of the coordinate expression for the vector product, is the so called "bac-cab" formula: a x (b x c) b(a' c) c(a'b) The scalar and vector products give rise to a number of similar identities that can be verified by checking the coordinate expressions of both sides of the equation. Below we list just a couple of these: a' (b x c) b' (c x a) c' (a x b), (a x b) ' (c x d) (a. c)(b.d) (".d)(b'c). Vector Calculus Consider a vector r which is the position vector for a particle P (see the figure below). Suppose
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This note was uploaded on 09/18/2011 for the course ASE 366K taught by Professor Lightsey during the Fall '08 term at University of Texas at Austin.

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ASE 366K - Vector Calculus Primer - 4 20 A ppendix. Vectors...

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