Calculus 3

Find the volume under the surface $z = 9 - x^2 - y^2$ and above the $xy$-plane.

Calculate the triple integral of the given region using spherical coordinates, where the region is bounded by a cone and a sphere.

Integrate the region described in spherical coordinates where $\rho = 1$, $0 \leq \phi \leq \frac{\pi}{6}$, and $0 \leq \theta \leq \frac{3\pi}{2}$.

Compute the curl of a given vector field $\vec{F}$.

Calculate the curl of the given vector field using the definition of the curl as the cross product of the del operator with the vector field.

Using the div, grad, and curl operators, solve a problem involving vector fields and partial differential equations.

Find the curl and divergence of the given vector field $\mathbf{F}(x, y, z) = (e^x \sin y, e^y \sin z, e^z \sin x)$.

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.

Calculate the surface integral of the vector field $F = \langle xy, yz, zx \rangle$ over the surface of a triangular prism with the given boundaries, using the Divergence Theorem.

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

Find the divergence of the vector field $F$ at the point (2, 4, 1).

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.

Maximize the function $f(x, y) = x^2 y$ subject to the constraint $x^2 + y^2 = 1$.

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.

Given the function $f(x, y)$ and the constraint $g(x, y) = 0$, use the Lagrange multipliers method to find the points at which $f(x, y)$ is maximized or minimized, with the specific example of $x^2 + y^2 = 100$.

Find the point on the circle $x^2 + y^2 = 4$ that is closest to the point $(3,4)$. Use the Lagrange multipliers method to solve.

Using the Cobb-Douglas production function $32L^{0.6}K^{0.4}$, maximize production subject to the constraint $4L + 2K = 15$.

Using the Cobb-Douglas production function $32L^{0.6} K^{0.4}$, maximize production subject to the constraint $4L + 2K = 15$.