MATH 215 Study Guide - Midterm Guide: Tangent Space, Divergence Theorem, Parametric Surface
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Consult the review sheets for the midterms for reviews of chapters 12 15. A vector eld on r2 is a function f from r2 to v2. Output: a vector f(x, y) = p (x, y)i + q(x, y)j. A graph of f can be obtained by plotting several vectors f(x, y) with initial point (x, y). A vector eld on r3 is a function f from r3 to v3. Output: a vector f(x, y, z) = p (x, y, z)i + q(x, y, z)j + r(x, y, z)k. Let c be a smooth curve in r2, parametrized by the vector-valued function r(t) = (cid:104)x(t), y(t)(cid:105), a t b. Let f be a continuous real-valued function of two variables. Line integral of f with respect to arc length: (cid:90) (cid:90) b f(x, y) ds = C t=a f(x(t), y(t))(cid:112)[x(cid:48)(t)]2 + [y(cid:48)(t)]2 dt (cid:90) b f(x(t), y(t))x(cid:48)(t) dt. Line integral of f with respect to y: f(x, y) dx =
