Calculus 3

Using Green's Theorem in its divergence form, calculate the outward flux across a given curve.

Using Stokes' Theorem, determine the counterclockwise circulation around a curve for a given surface.

Apply the fundamental theorem of line integrals to measure the flow along a curve when the vector field can be written as the gradient of a function.

Using the divergence concept, determine if a vector field at a given point has a positive, negative, or zero divergence.

Using the Divergence Theorem, calculate the outward flux of a vector field across a closed, smooth surface, given that the field is defined over a three-dimensional vector space with components M, N, and P.

Find the maximum or minimum of a function $f(x, y)$ subject to a constraint $g(x, y) = k$ using the method of Lagrange multipliers.

Compute the surface area of a surface given its parametric description. Use the formula: $\ iint_{\text{Region}} \| \mathbf{r}_u \times \mathbf{r}_v \| \, du \, dv$ where $\mathbf{r}_u$ and $\mathbf{r}_v$ are the partial derivatives of the position vector with respect to the parameters $u$ and $v$.

Compute the line integral of the vector field F on a curve $C$, using the parameterization $\mathbf{r}(t)$ from $t = a$ to $t = b$. The line integral is given by $\int_{a}^{b} \mathbf{F}(\mathbf{r}(t)) \cdot \frac{d\mathbf{r}}{dt} \, dt$.