1 partial derivatives

1.1 linear approximation

\Delta z = f_x(x_0, y_0)\Delta x + f_y(x_0, y_0)\Delta y

1.2 chain rule

Let F(t) = f(g(t), h(t), i(t)) then:

F'(t) = f_x(g(t), h(t), i(t))g'(t) + f_y(g(t), h(t), i(t))h'(t) + f_z(g(t), h(t), i(t))i'(t)

1.2.1 matrix function

\frac{\partial(u_1, \cdots ,u_m)}{\partial (x_1, \cdots, x_n)} = \begin{pmatrix} \frac{\partial u_1}{\partial x_1} & \frac{\partial u_1}{\partial x_2} & \cdots & \frac{\partial u_1}{\partial x_n} \\ \frac{\partial u_2}{\partial x_1} & \frac{\partial u_2}{\partial x_2} & \cdots & \frac{\partial u_2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots\\ \frac{\partial u_m}{\partial x_1} & \frac{\partial u_m}{\partial x_2} & \cdots & \frac{\partial u_m}{\partial x_n} \\ \end{pmatrix}

1.2.2 generalization

\frac{\partial(u_1,\cdots,u_m)}{\partial (t_1,\cdots,t_k)} = \frac{\partial(u_1,\cdots,u_m)}{\partial (x_1,\cdots,x_n)} \frac{\partial(x_1,\cdots,x_n)}{\partial (t_1,\cdots,t_k)}

1.3 constrained extrema

To find extrema of a function in a region we first find the extrema inside the region with 2nd derivate test and then separately find extrema on the boundary.

1.3.1 Lagrange multipliers

To find the extreme values of f(x,y) on a g(x,y) = c constraint solve the system of equations:

\begin{cases} f_x(x,y) = \lambda g_x(x,y)\\ f_y(x,y) = \lambda g_y(x,y)\\ g(x,y) = c \\ \end{cases}

1.4 Gaussian integral

\int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi}