Calculus 3 — Paul Notes

If region $D$ is bounded by curves:

In Calc 3, we extend the 2D plane ($xy$) to 3D space ($xyz$).

Relates a line integral around a simple closed curve $C$ to a double integral over the region $D$ inside $C$. $$\oint_C P , dx + Q , dy = \iint_D \left( \frac\partial Q\partial x - \frac\partial P\partial y \right) , dA$$

: This chapter focuses on optimization, including finding relative extrema and using Lagrange Multipliers to solve constrained optimization problems.

If transforming from $xy$ to $uv$: $$\iint_R f(x,y) , dA = \iint_S f(x(u,v), y(u,v)) \left| \frac\partial(x,y)\partial(u,v) \right| , du , dv$$ The Jacobian determinant is: $$\frac\partial(x,y)\partial(u,v) = \beginvmatrix \frac\partial x\partial u & \frac\partial x\partial v \ \frac\partial y\partial u & \frac\partial y\partial v \endvmatrix$$