Multivariate calculus problem: Calculating the gradient vector

[squenshl]

Homework Statement

Let $U := \left\{(x,y)\in \mathbb{R}^2: xy\neq 0\right\}$ and let $f: U\mapsto \mathbb{R}$ be defined by
$$f(x,y) := (\log_{e}{(|x|)})^2+(\log_{e}{(|y|)})^2.$$


1. Calculate $\nabla f(x,y)$ at each point of $U$.

2. Let $\mathbf{r}: (0,1)\mapsto \mathbb{R}^2$ be defined by $\mathbf{r}(t) := \left(e^{\sin{(t)}},e^{\cos{(t)}}\right).$
Calculate the derivative of $\mathbf{r}$ at each point of $(0,1).$

3. Justify whether you can use the chain rule to calculate the derivative of $f\circ \mathbf{r}.$
If it is justifiable, calculate the derivative of $f\circ \mathbf{r}$ using the chain rule.

Relevant Equations

None

1. We find the partial derivatives of $f$ with respect to $x$ and $y$ to get $f_x = \frac{2\ln{(x)}}{x}$ and $f_y = \frac{2\ln{(y)}}{y}.$ This makes the gradient vector
$$\nabla{f} = \begin{bmatrix}
f_x \\
f_y
\end{bmatrix} = \begin{bmatrix}
\frac{2\ln{(x)}}{x} \\
\frac{2\ln{(y)}}{y}
\end{bmatrix}.$$

2. We have
$$\mathbf{r}'(t) = \left(\cos{(t)}e^{\sin{(t)}},-\sin{(t)}e^{\cos{(t)}}\right).$$

After this I'm a little confused. Any help is appreciated.

 

[member 587159]

For (3), what does your version of the chain rule say?

If your multivariable calculus textbook is rigorous, it might also want you to show that $U$ is an open set (which is easily seen by a drawing).

 

[squenshl]

The derivative of $\mathbf{r}$ at each point of $(0,1)$?

 

[vela]

The derivative of $\mathbf{r}$ at each point of $(0,1)$?

That confused me initially too. It would be clearer if it said "at each point in the open interval (0,1)."

 

[squenshl]

Okay (1) and (2) are done.
So for (3), assuming $t > 0$, $f\circ \mathbf{r} = \ln{(e^{\sin{(t)}})}^2+\ln{(e^{\sin{(t)}})}^2 = \sin^2{(t)}+\cos^2{(t)} = 1$ so the derivative is $0$.

 

[vela]

For #3, you need calculate the derivative using the chain rule if it can be applied.

 

[squenshl]

Okay then I’m lost how do we then justify whether to use the chain rule?

 

[vela]

I'm sure the conditions are stated in your textbook or were covered in lecture. Look them up.