Calculus 3

Sketch the vector field $\mathbf{F}(x, y) = x\hat{i} + y\hat{j}$ and describe the starburst pattern that you observe.

Find the force vector at a given point (x, y) in a 2D rotational vector field given by $F(x, y) = (y, -x)$.

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.

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 $\int_{a}^{b} \mathbf{F}(\mathbf{r}(t)) \cdot \frac{d\mathbf{r}}{dt} \, dt$.

Evaluate the integral $\int_{0}^{\frac{\pi}{3}} \sin(x) \cos\left(\frac{\pi}{3} - x\right) \, dx$.

Calculate the volume of a truncated wedge with dimensions: 2 units high, 5 units at the end, 6 units long, and 4 units wide, using a triple integral in rectangular coordinates.

Evaluate the triple integral from 0 to $\pi$ and 0 to 2 and then 0 to $\sqrt{4 - Z^2}$ of $Z \sin Y$ with respect to $dX$ $dZ$ $dy$.

Evaluate the triple integral: $\int_{2}^{4} \int_{-1}^{1} \int_{0}^{3} (y - xz) \, dz \, dy \, dx$.