Download MATH 218 Final Exam Prep: Vector & Multivariable Calculus and more Exams Mathematics in PDF only on Docsity!
MATH 218 Final Exam Prep with
Questions and Answers
Linearization formula ANSWER L = integral from a to b of f(g(t), h(t), k(t)) |v(t)| dt gradient field ANSWER delta g = (dg/dx)i + (dg/dy)j + (dg/dz)k work equation ANSWER W = integral from a to b of (Mdx + Ndy + Pdz) dt Flux equation ANSWER Flux = integral of M1dy - N1dx + integral of M2dy - N2dx (both w.r.t. t) how to tell if conservative ANSWER conservative if dM/dy = dN/dx, dM/dz = dP/dx, and dN/dz = dP/dy greens theorem flux ANSWER flux = double integral over R of (dN/dx + dM/dy) dxdy greens theorem circulation ANSWER circulation = double integral over R of (dN/dx - dM/dy) xydy surface area of explicit ANSWER |S| = double integral of sqrt (1 + (fx)^2 + (fy)^2) dxdy surface area of implicit ANSWER |S| = double integral of | gradient F | / | gradient f dot p | AKA length over abs value surface area of parametric ANSWER |S| = |ru x rv| dudv WHERE ru is derivative w.r.t. u and ...
|S| = ANSWER double integral of 1 dsigma when the 1 in |S| is not 1 ANSWER replace it with G(x, y, f(x, y))dsigma OR G(x,y,z)dsigma OR G(f(u,v), g(u,v) h(u,v)dsigma Flux of |S| ANSWER Flux = double integral of F dot n dsigma For explicit, n = +- <fx, fy, -1> / sqrt (1 + (fx)^2 + (fy)^2) for implicit, n = +- gradient g / | gradient g | for parametric, n = +- ru x rv / | ru x rv | curl equation ANSWER curl = dell x F = (matrix from 1-9) i, j, k, d/dx, d/dy, d/dz, M, N, P (Remember to alternate + - +) circulation (found with curl) ANSWER circulation (integral of F dot Tds) = flux (double integral over S of curl dot n dsigma) RMK: n can be i, j, k, or something else RMK: use a FLAT surface for S some other theorem ANSWER flux (double integral over S of F dot n dsigma) = divergence integral (triple integral over D of dell x F dV) WHERE dell x F = dM/dx + dN/dy + dP/dz length of a curve formula ANSWER Length = integral from a to b over |v(t)| dt remember that ANSWER changing the order of integration can be useful
