# apply chain rule to get this result

Given $$g(\tau_e, t, \vec x, \vec \zeta) \equiv \tau_e - t + \frac{1}{c_0} |\vec x - \vec y(\vec \zeta, \tau_e)| = 0$$

The textbook "Aeroacoustics" said applying for chain rule, will have $$\left(\frac{\partial g}{\partial x_j}\right)_{\tau_e=const} + \left(\frac{\partial g}{\partial \tau_e}\right)_{\vec x=const}\frac{\partial \tau_e}{\partial x_j} =0$$

How did it come?

• You can get parentheses (and other paired delimiters) to adjust to the size of their content by preceding them with \left and \right. Commented Jul 14, 2016 at 19:41
• What's being held constant in $\frac{\partial\tau_e}{\partial x_j}$? Commented Jul 14, 2016 at 19:42
• Also, what do you mean by $\left(\frac{\partial g}{\partial x_j}\right)_{\tau_e=\text{const}}$ when $g$ depends on further variables beyond $\vec x$ and $\tau_e$? Commented Jul 14, 2016 at 19:43
• @joriki, thanks for the comments! For your second, question, it was not written in the textbook...I copied exactly what has been said in the textbook to here. Commented Jul 14, 2016 at 19:52

assuming $$\vec y = \text{const}$$ $$\vec \zeta = \text{const}$$
so consider $\tau_e$ as a function of $x_j$, the result is the got.