Unit Vectors in 3D

Question

Hello everyone I am having trouble understanding the question, specifically the $\vec{e}_r$ part, I don't know what it wants me to do\

The the gradient operator in spherical coordinates is this.

So I got:

$$<e^{\sin \left(x\right)}\cos \left(x\right), \sqrt{x^2+y^2+z^2}, xyz>$$

Is this the answer they want? Or are they asking for something completely different, because I'm really confused reading the problem and don't understand

Edit:

So here is the final correct answer: $$<e^{\sin \left(x\right)}\cos \left(x\right), 0, 0>$$

Answer 1

There is something wrong with you computation. The radius $r = r(x,y,z) = \sqrt{x^2+y^2+z^2}$. Do you see now what your error was?

$\textbf{Edit}$: Also, remember that $e_r$ is just fancy notation for:

$$\frac{\langle x,y,z\rangle}{\sqrt{x^2+y^2+z^2}}$$

Answer 2

Use $\frac{\delta r}{\delta x}=\frac{x}{r}$,$\frac{\delta r}{\delta y}=\frac{y}{r}$ and $\frac{\delta r}{\delta z}=\frac{z}{r}$.

Then grad$\phi=e^{sin r}cos r(\hat i\frac{\delta r}{\delta x}+\hat j\frac{\delta r}{\delta y}+\hat k\frac{\delta r}{\delta y})$

$=e^{sinr}$$cos r$$\overrightarrow e_r$