Differentiation through product of variables

Question

I am working on a problem connected to shallow water waves. I have a vector:

$U = \begin{bmatrix} h \\ h \cdot v_1\\ h \cdot v_2\end{bmatrix}$

and a function

$f(U) = \begin{bmatrix} h \cdot v_1 \\ h \cdot v_1^2 + 0.5\cdot gh^2\\ h \cdot v_1 \cdot v_2\end{bmatrix}$

I now want to calculate the Jacobian Matrix of $f(U)$.

However, I am lost at how to calculate the partial derivatives when it comes to differentiating through a product. I.e.:

$\frac{\partial (h \cdot v_1^2 + 0.5 \cdot gh^2)}{\partial (h \cdot v_1)} = \frac{\partial (h \cdot v_1^2) }{\partial (h \cdot v_1)} + \frac{\partial (0.5 \cdot gh^2)}{\partial (h \cdot v_1)} = v_1 + ? \dots$
or

$\frac{\partial (h \cdot v_1^2 + 0.5 \cdot gh^2)}{\partial (h \cdot v_2)} = \frac{\partial (h \cdot v_1^2) }{\partial (h \cdot v_2)} + \frac{\partial (0.5 \cdot gh^2)}{\partial (h \cdot v_2)} = \dots$

Googling it is really difficult and brought no result, since I only ever find explanations for the product rule... Maybe someone here could enlighten me! Any kind of pointer in the right direction is highly appreciated! Thanks so much in advance!

Answer 1

In the final analysis your $f$ is a function $f:\>{\mathbb R}^3\to{\mathbb R}^3$ taking the variables $h$, $v_1$, $v_2$ as input and producing three scalar values $$u:=hv_1,\quad v:=hv_1^2+{g\over2}h^2,\quad w:=hv_1v_2$$ as output, whereby $g$ seems to be some constant. By definition the Jacobian of $f$ is the matrix of partial derivatives $$\left[\matrix{u_h& u_{v_1}&u_{v_2}\cr v_h& v_{v_1}&v_{v_2}\cr w_h& w_{v_1}&w_{v_2}\cr}\right]=\left[\matrix{v_1& h&0\cr v_1^2+gh& 2v_1h&0\cr v_1v_2&hv_2&hv_1\cr}\right]\ .$$

Answer 2

To do this, simply get rid of the "products of variables" by a substitution.

First, set $w_1 = h v_1$ and $w_2 = h v_2$.

Next, solve for $v_1 = w_1 / h$ and $v_2 = w_2 / h$.

Next, substitute for $v_1,v_2$ and simplify:

$U = \begin{bmatrix} h \\ w_1\\ w_2\end{bmatrix}$

$f(U) = \begin{bmatrix} w_1 \\ w_1^2 \bigm/ h + 0.5\cdot gh^2\\ w_1w_2 \bigm/ h \end{bmatrix}$

Next, compute the Jacobian Matrix for $f(U)$ in the usual way.

Finally, substitute for $w_1$ and $w_2$ and simplify.

Remarks: In your comment you ask about the "very first" entry of the Jacobian matrix.

The meaning of partial derivatives depends on the full set of coordinate variables, not just on a single variable.

In your problem, you have informed us that the coordinate variables are $h$, $hv_1$, $hv_2$. I have simply subsituted these with one letter symbols $h$, $w_1$, $w_2$, respectively.

The meaning of the partial derivative $\frac{\partial w_1}{\partial h}$, using the coordinate variables you specified, means that you hold $w_1$ and $w_2$ constant and, while holding them constant, you vary $h$ and take the derivative of $w_1$ with respect to $h$. The result is zero, because $w_1$ has been held constant.

You can translate this back into your own notation: the meaning of $\frac{\partial hv_1}{\partial h}$ is that you hold $hv_1$ and $hv_2$ constant and, while holding them constant, you take the derivative of $hv_1$ with respect to $h$. Since $hv_1$ is held constant, its derivative is zero.

In other words, the definition of the partial derivative $\frac{\partial}{\partial h}$ is dependent on which other two coordinates you choose: its definition with coordinates $w_1,w_2$ is not the same as its definition with coordinates $v_1,v_2$. You can verify this for yourself if you look up the actual definition of partial derivatives, as a limit of difference quotients. See here for a discussion.