# UNSW MATH2069 Complex Analysis and Vector Calculus

## Vector Calculus

$$\int_{Birth}^{Death} Study \cdot dt = Life$$

Directional derivative

$$\nabla f_{(x,y)} \cdot u$$

Minimum and Maximum

$$D = f_{xx}f_{yy} – (f_{xy})^2$$

 D > 0 $$f_{xx}$$ < 0 local maximum D > 0 $$f_{xx}$$ > 0 local minimum D < 0 saddle point D = 0 inconclusive

Differential

$$\Delta f = f_x \,dx + f_y \,dy$$

Arc Length

$$L = \int_{a}^{b} ||r\prime(t)||dt = \int_{a}^{b} \sqrt{ (\frac{dx}{dt})^2+(\frac{dy}{dt})^2 +(\frac{dz}{dt})^2}$$

Change of Varible

Sphere perimatisation

$$r(\theta,\phi) = (R \cos \theta \sin \phi, R \sin \theta \sin \phi, R \cos \phi)$$

Green’s Theorem
$$\oint_{C} (L\, dx + M\, dy) = \iint_{D} \left(\frac{\partial M}{\partial x} – \frac{\partial L}{\partial y}\right)\, dx\, dy$$

Divergence Theorem

$$\int_S \mathrm{div}_S(\textbf{F})\,dA \;=\; \oint_{\partial S} \textbf{t}\cdot(\textbf{n}\times\textbf{F})\; ds.$$

Line integral

$$\int_{a}^{b} F(r(t)) \cdot r \prime (t) \,dt$$

### Tips and Tricks

Integral of Sine squared

$$\int \sin^2 x \,dx = \int \frac{1 – \cos 2x}{2} \,dx = \frac{x}{2} – \frac{1}{4} \sin 2x + C$$

Integral of Cosine squared

$$\int \cos^2 x \,dx = \int \frac{1 + \cos 2x}{2} \,dx = \frac{x}{2} + \frac{1}{4} \sin 2x + C$$