Find the value of constant 'a' such that $\mathbf{F} = (x + 3y)i + (y - 2z)j + (x + az)k$ is solenoidal.
abla $\cdot \mathbf{F} = 1 + 1 + a = 0 => 2 + a = 0 => a = -2$.
If $\mathbf{r} = x i + y j + z k$ and r = |$\mathbf{r}|$, then
abla $\cdot \mathbf{r}$ is equal to:
abla $\cdot (xi + yj + zk) = 1 + 1 + 1 = 3$.
If $\mathbf{r} = x i + y j + z k, what$ is the magnitude of
abla $\times \mathbf{r}$?
abla $\times (xi + yj + zk) = 0i + 0j + 0k$. Its magnitude is 0.
According to Gauss's Divergence Theorem, $\iint_S \mathbf{F} \cdot d\mathbf{S}$ is equal to:
Gauss's Divergence Theorem converts a closed surface integral into a volume integral over the enclosed region.
According to Stokes' Theorem, $\oint_C \mathbf{F} \cdot d\mathbf{r}$ is equal to:
Stokes' Theorem relates the line integral around a closed curve C to the surface integral of curl F over a surface S bounded by C.
