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If a function $f(t,x)$ has $x \in \mathbb{R}^{2}$, what is the partial derivative $\frac{\partial{f}}{\partial{x}}$? Thank you greatly.

user314849
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  • There is more than one form of derivative for functions of several variables. Look up divergence, gradient and curl. – Paul Oct 14 '18 at 13:04

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In such context, $\partial f/\partial x$ can simply denote $(\partial f/\partial x_1, \partial f/\partial x_2)$, where $x=(x_1,x_2)$. Probably this is the most common meaning of it.

Rodrigo Dias
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  • I thought it could be a Jacobian, since the question is if $f \in C^{1}$. – user314849 Oct 14 '18 at 13:19
  • The notation $(\partial f/\partial x_1,\partial f/\partial x_2)$ is used for Jacobian as well, since it equals $\begin{pmatrix} \partial f_1/\partial x_1 & \partial f_1/\partial x_2 \ \partial f_2/\partial x_1 & \partial f_2/\partial x_2 \end{pmatrix}$ for $f=(f_1,f_2)$. – Rodrigo Dias Oct 14 '18 at 13:36