Cheat Sheet.pdf, Summaries of Calculus

Download Cheat Sheet.pdf and more Summaries Calculus in PDF only on Docsity! Math 2110Q: Helpful Formulas 1. The Second Derivative Test Let pa, bq be a critical point of a function fpx, yq with Dpx, yq “ fxxfyy ´ f2 xy. 1. If Dpa, bq ą 0, then pa, bq is either a local maximum or minimum (a) fxxpa, bq ă 0 ñ pa, bq is a local maximum (b) fxxpa, bq ą 0 ñ pa, bq is a local minimum 2. Dpa, bq ă 0 ñ pa, bq is a saddle 3. Dpa, bq “ 0 ñ the test is inconclusive 2. Summary of Line Integrals and Surface Integrals LINE INTEGRALS SURFACE INTEGRALS C : ~rptq, a ď t ď b S : ~rpu, vq, pu, vq P D ds “ |~r 1ptq| dt “arc length differential dS “ |~ru ˆ ~rv| dA= surface area differential ż C fpx, y, zq ds “ ż b a fp~rptqq|~r 1ptq| dt ij S fpx, y, zq dS “ ij D fp~rpu, vqq|~ru ˆ ~rv| dA (independent of orientation of C) (independent of orientation of S) d~r “ ~r 1ptq dt d~S “ p~ru ˆ ~rvq dA ż C ~F ¨ d~r “ ż b a ~F p~rptqq ¨ ~r 1ptq dt ij S ~F ¨ d~S “ ij D ~F p~rpu, vqq ¨ p~ru ˆ ~rvq dA (depends on orientation of C) (depends on orientation of S) Theorems that may apply: Theorems that may apply: Fundamental Theorem for Line Integrals Stokes’ Theorem Green’s Theorem Divergence Theorem