Surface Integrals, Divergence, Curl and Potential
Exercise 1 : Coordinate Planes
Given three squares with an area of 4 units each. They are placed:
- in the x-yplane,
- in the x-zplane, and
- in the y-zplane.
Determine the surface elements.
Exercise 2 : Rectangle Vector Element
Given a rectangle with area $a \cdot b$, determine the vector element.
Exercise 3 : Flux Through Given Surface Elements
Compute the flow of the vector field $\mathbf{F}(x, y, z) = (5, 3, 0)$ through the surfaces given by the respective surface elements:
- $\mathbf{A} = (1, 1, 1)$
- $\mathbf{A} = (2, 0, 0)$
- $\mathbf{A} = (0, 3, 1)$
Exercise 4 : Flux Through a Sphere (Constant Field)
Compute the flow of the vector field $\mathbf{F}(x, y, z) = (2, 2, 4)$ through a sphere centered at the origin with radius $R = 3$.
Exercise 5 : Flux Through Spheres — Radial Fields
Compute the flow of the vector fields through a sphere centered at the origin with radius $R$.
- $\mathbf{F}(x, y, z) = \frac{3(x, y, z)}{x^2 + y^2 + z^2}$
- $\mathbf{F}(x, y, z) = \frac{(x, y, z)}{\sqrt{1 + x^3 + y^3 + z^3}}$
Exercise 6 : Divergence Theorem (Cube)
Let $\mathbf{F}(x,y,z) = (x^2, y^2, z^2)$. Compute the outward flux of $\mathbf{F}$ through the boundary of the cube $0 \le x,y,z \le 1$ using the divergence theorem.
Exercise 7 : Stokes’ Theorem — Circulation over a Disk
Let $\mathbf{F}(x,y,z) = (-y, x, 0)$. Compute the circulation $\oint_{\partial D} \mathbf{F}\cdot d\mathbf{r}$ around the circle $x^2 + y^2 = 1$ in the $z=0$ plane by applying Stokes’ theorem to the unit disk $D$.
Exercise 8 : Conservative Field and Potential
Determine whether the vector field $\mathbf{F}(x,y,z) = (2xy, x^2 + 2z, y^2)$ is conservative in a simply connected region. If it is, find a scalar potential function $\phi(x,y,z)$ such that $\nabla \phi = \mathbf{F}$.
Exercise 9 : Surface Integral over a Graph
Compute the surface integral $\iint_S \mathbf{F}\cdot d\mathbf{S}$ for $\mathbf{F}(x,y,z) = (0,0,z)$ where $S$ is the graph $z = x^2 + y^2$ over the disk $x^2 + y^2 \le 1$, with upward orientation.
Exercise 10 : Flux Through a Cylinder (Lateral Surface)
Let $\mathbf{F}(x,y,z) = (x, y, z)$ and let $S$ be the lateral surface of the cylinder $x^2 + y^2 = R^2$, $0 \le z \le h$, oriented outward. Compute the flux through $S$.
Exercise 11 : Parametrized Surface and Unit Normal
Parametrize the portion of the paraboloid $z = 4 - x^2 - y^2$ above the plane $z=0$ and compute the unit normal vector field $\hat{n}$ pointing upward. Then evaluate $\iint_S \hat{n}, dS$ (i.e., the vector area) for this surface.