# 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. – joriki Jul 14 '16 at 19:41
• What's being held constant in $\frac{\partial\tau_e}{\partial x_j}$? – joriki Jul 14 '16 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 $\tau_e$? – joriki Jul 14 '16 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. – ArtificiallyIntelligence Jul 14 '16 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.