Calculus 3: Vector fields, divergence, and curl

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 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).

By hand, draw the vector field $\mathbf{F}(x, y) = y\hat{i} - x\hat{j}$ and plot various points to visualize the pattern.

For a vector field given by components $f(x, y) = (y, x)$, sketch the vector field.

Given a gravitational field, represented as $F = -\frac{G m_1 m_2}{r^2} \hat{r}$, where $G$ is the gravitational constant, $m_1$ and $m_2$ are masses, and $r$ is the distance, derive the expression for the gravitational field as a vector field.