Vector Calculus Unit 3 Quiz

The Gauss theorem, also known as Gauss's divergence theorem, states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the field over the region enclosed by the surface.

The gradient of a scalar field $f(x, y, z)$ is a vector field denoted by $\nabla f$, where $\nabla = \frac{\partial}{\partial x} \hat{i} + \frac{\partial}{\partial y} \hat{j} + \frac{\partial}{\partial z} \hat{k}$, and its components are the partial derivatives of $f$ with respect to $x$, $y$, and $z$.

The divergence of a vector field $\vec{F}$ is a scalar function denoted by $\nabla \cdot \vec{F}$, and it represents the volume density of the outward flux of $\vec{F}$ per unit volume at a given point.

A force $\vec{F}$ is said to be conservative if it is the gradient of a scalar potential function. Mathematically, this means that $\vec{F}$ is conservative if $\vec{F} = -\nabla V$, where $V$ is the potential function.

The curl of $\vec{A}$ is given by $\nabla \times \vec{A} = (\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z})\hat{i} - (\frac{\partial A_z}{\partial x} - \frac{\partial A_x}{\partial z})\hat{j} + (\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y})\hat{k}$. Evaluating this at $(2, 3, 4)$ gives the curl of $\vec{A}$ at that point.