Mathematical Methods in Physics Quiz

$\nabla f = (6x, 2z, 2y)$

$\nabla \bullet F = 6x + 2z$

$\nabla \times G = (0, 0, 1)$

The formula for the gradient of a scalar function $f(x, y, z)$ is given by $\nabla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k$.

The divergence of a vector function $\mathbf{F} = (P, Q, R)$ is defined as $\nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z$.

The formula for the curl of a vector function $\mathbf{G} = (M, N, P)$ is given by $\nabla \times \mathbf{G} = \left(\frac{\partial P}{\partial y} - \frac{\partial N}{\partial z}\right)\mathbf{i} - \left(\frac{\partial P}{\partial x} - \frac{\partial M}{\partial z}\right)\mathbf{j} + \left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\right)\mathbf{k$.

The Gauss Divergence Theorem states that the flux of a vector field $\mathbf{F}$ through a closed surface is equal to the volume integral of the divergence of $\mathbf{F}$ over the region enclosed by the surface. Mathematically, it can be expressed as $\iint_{S} \mathbf{F} \cdot d\mathbf{S} = \iiint_{V} (\nabla \cdot \mathbf{F}) dV$, where $S$ is the closed surface and $V$ is the region enclosed by the surface. This theorem is applied to relate surface integrals to volume integrals in vector calculus.

In vector calculus, line integrals, surface integrals, and volume integrals are related through the fundamental theorem of calculus and the Gauss Divergence Theorem. Line integrals are used to calculate the work done by a force field along a curve, surface integrals are used to calculate flux through a surface, and volume integrals are used to calculate properties over a three-dimensional region. These integrals are interconnected through the relationships established by the theorems in vector calculus.