Variables:
k is a constant vector where k = <k1,k2,k3>
A is a constant vector
r is a position vector where r = <x,y,z>
w is a constant scalar
v = Asin(k·r - wt)
Problem:
show that: ∇·v = k·Acos(k·r - wt)
*looking at the left side of the equation
∇·Asin(k·r - wt)
k·r = x(k1) + y(k2) + z(k3)
*using product of a scalar and vector rule
sin(x(k1) + y(k2) + z(k3) - wt)(∇·A) + A·(∇sin(x(k1) + y(k2) + z(k3) - wt)
This is what I have so far. I'm confused on what I should do (well a bit in general) but especially with the second part of the last line.
Should the gradient be calculated using the partial derivatives with respect to k1,k2,k3 or should it be done with respect to x,y,z? Should it be done with neither?
I'm guessing x,y,z since k is a constant vector but that's just a hunch and I'd like to know the real explanation if that is correct.
Thanks.