If \phi(x, y, z) = $x^2$ y + $y^2$ z + $z^2$ x, find
abla $\phi$ at the point (1, 1, 1). Magnitude of
abla $\phi =$?
abla $\phi = (2xy + z^2)i + (x^2 + 2yz)j + (y^2 + 2zx)k$. At (1,1,1),
abla $\phi = 3i + 3j + 3k$. Magnitude = $\sqrt{3^2 + 3^2 + 3^2} = \sqrt{27} \approx 5.196 \approx 5.2$.
A vector field $\mathbf{F}$ is called Solenoidal if:
A vector field is solenoidal if its divergence is zero everywhere (
abla $\cdot \mathbf{F} = 0)$.
A vector field $\mathbf{F}$ is Irrotational if:
A vector field is irrotational (conservative) if its curl is zero everywhere (
abla $\times \mathbf{F} = \mathbf{0})$.
Find the directional derivative of $\phi(x,y,z) = x^2 y z + 4xz^2$ at (1, -1, 1) in the direction of $\mathbf{v} = 2i - j - 2k$.
abla $\phi(1,-1,1) = 2i + j + 7k$. Direction unit vector u = (2i - j - 2k)/3. Directional derivative = (4 - 1 - 14)/3 = -11/3 = -3.67.
If \mathbf{F} = $x^2$ i + $y^2$ j + $z^2$ k, compute
abla $\cdot \mathbf{F}$ at the point (1, 2, 3).
abla $\cdot \mathbf{F} = \frac{\partial}{\partial x}(x^2) + \frac{\partial}{\partial y}(y^2) + \frac{\partial}{\partial z}(z^2) = 2x + 2y + 2z$. At (1,2,3): 2(1) + 2(2) + 2(3) = 2 + 4 + 6 = 12.
