Calculate the gradient vector of the following scalar field:
$\Phi(r)=\cos(ar)$ where $r$ is the 3-dimensional position (vector) and $a$ is a constant vector.
So gradient means that I need to partially derivative the given function, right? And the components of the gradient are the partial derivatives. A vector can be given like this in 3 dimension: $$v=(ax, by, cz)$$ where $a$, $b$, $c$ are the coefficients, but what is the case with this exercise? Can someone please explain me, how can I solve it?
Best Answer
Let $\partial_l=\frac{\partial}{\partial x_l}$, $a=(a_1,a_2,a_3);r=(r_1,r_2,r_3)$
$$\nabla(\Phi)=\begin{pmatrix}\partial_1\cos(a_1r_1+a_2r_2+a_3r_3)\\ \partial_2\cos(a_1r_1+a_2r_2+a_3r_3)\\ \partial_3\cos(a_1r_1+a_2r_2+a_3r_3)\end{pmatrix}=\begin{pmatrix} -a_1\sin(a\cdot r)\\ -a_2\sin(a\cdot r)\\ -a_3\sin(a\cdot r)\end{pmatrix}$$