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2009-11-10

**Last Modified:**
2009-11-10

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SUMMARY: topmatter (slides 1-6) def'n and examples of fractional error (slide 7) rotationally invariant symmetric matrices are scalar (slide 8) def'n of scalar quadratic forms and diagonal quadratic forms (slide 9) Homework assignment (04/23)-1 (slides 10-12) spherical coordinates, parameterizing (almost all of) a ball; SKILL: Use change of variables formula to find areas/volumes of well parametrized sets (slide 13) the Cantor set, def'n and properties (slides 14-25) example of a strictly increasing (continuously differentiable) function whose derivative vanishes uncountably often (slides 26-27) the change of variables formula and the idea of its proof (slides 28-41) the change of variables formula for polar coordinates (slides 42-47) derivation of the integral, from -\infty to \infty, of e^{-x^2/2} dx (slides 48-51) start of Green's Theorem and Cauchy's Theorem (slide 52) def'ns: directed line segment, starting point, ending point (slide 53) def'ns: standard parametrization, constant velocity (slide 54) def'n and illustration: simple chain (slide 55) def'ns and examples: rectangle, open rectangle (slides 56-57) notation and examples: counterclockwise boundary, denoted \partial R (slides 58-59) formal def'n: counterclockwise boundary, denoted \partial R (slide 60) def'n of line integral of a one-form over a directed line segment (slide 61) def'n of line integral of a one-form over a simple chain (slide 62) first statement of Green's Theorem (slide 63) def'ns: zero-form, one-form, exterior derivative of a zero-form, notation dF (slide 64) SKILL: Compute the exterior derivative of a zero-form (slide 65) def'n of a two-form, conventions for the algebraic manipulation of forms, antisymmetry of \wedge on one-forms (slide 66) f(A\wedge B)=(fA)\wedge B=A\wedge(fB), i.e., zero-forms move around through forms with impunity (slide 67) SKILL: Algebraic collection of terms in two-forms (slide 68) def'n: exterior derivative of a one-form, denoted dF (slide 69) SKILL: Compute exterior derivatives of zero-forms and one-forms; example of computation of the exterior derivative of a zero-form (slide 70) example of computation of the exterior derivative of a one-form (slide 71) general computation of d(P dx + Q dy), the exterior derivative of a generic one-form; restatement of Green's Theorem using notation about forms (slide 72)