Multi-variable Differential Calculus

Partial Derivatives

Derivatives of multivariable functions that only take into account one variable at a time, while keeping the others constant

Let $z = f (x, y)$ be a function with mixed second partial derivatives that are continuous at a point $(a, b)$ , then

\frac{\partial^{2} f}{\partial x \partial y} = \frac{\partial^{2} f}{\partial y \partial x} at (a, b)

A function $f (x, y)$ is differentiable at $(a, b)$ if and only if

$f_{x} (a, b)$ and $f_{y} (a, b)$ exist
$lim_{\tilde{x} \to \tilde{a}} \frac{| R_{1, \tilde{a}} (\tilde{x}) |}{| | \tilde{x} - \tilde{a} | |}$

where $R_{1, \tilde{a}} (\tilde{x})$ denotes the remainder of the first order Taylor approximation of $f$ at $\tilde{a}$

\begin{aligned} R_{1, \tilde{a}} & = f (x, y) - T_{1, \tilde{a}} (\tilde{x}) \\ R_{1, \tilde{a}} & = f (x, y) - (f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)) \end{aligned}

and $| | \cdot | |$ denotes the Euclidean norm

If $f (x, y)$ is differentiable at $(a, b)$ then $f (x, y)$ is continuous at $(a, b)$

contrapositive also true: $f (x, y)$ not continuous at $(a, b) \to$ not differentiable at $(a, b)$

If the partial derivatives $f_{x}$ and $f_{y}$ exist near $(a, b)$ and are continuous at $(a, b)$ then $f$ is differentiable at $(a, b)$

f_{x} = \frac{\partial f}{\partial x} = l i m_{h \to 0} \frac{f (x + h, y) - f (x, y)}{h}

f_{y} = \frac{\partial f}{\partial y} = l i m_{h \to 0} \frac{f (x, y + h) - f (x, y)}{h}

If $z = f (x, y) = f (x (t), y (t))$ is a differentiable function with both $x$ and $y$ depending on a different variable

\frac{d f}{d t} = \frac{d z}{d t} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial t}