mystical madtopia

−b±b2−4ac2a\frac{-b \pm \sqrt{b^{2}-4ac}}{2a}

Quadratic Formula: It can find the roots of any quadratic function. Memorize easy

∫udv=uv−∫vdu\int_{}^{}udv = uv – \int_{}^{}vdu

Integration By Parts: Allows you to integrate two functions that are multiplied by each other. Tabular method for integrating by parts multiple times

∇f(x,y)=<δfδx,δfδy>=fxi+fyj\nabla f(x,y)=\lt \frac{\delta f}{\delta x}, \frac{\delta f}{\delta y}\gt = f_{x}i+f_{y}j

Gradient Vector: Points in the direction of steepest increase of a function. Geometric meaning

[y1y2y1′y2′]\begin{bmatrix} y_{1} & y_{2} \\ y’_{1} & y’_{2}{} \end{bmatrix}

The Wronskian: Used to determine if a set of solutions is linearly independent and forms a fundamental solution set. Learned this using the variation of parameters method for variable coefficient ordinary differential equations

a→⋅b→=|a→||b→|cos⁡θ\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos\theta

Dot Product: Amazing blog post and video

a→×b→\vec{a} \times \vec{b}

Cross Product: Operation that produces a third vector that is orthogonal to both vectors a and b

[x=1]x3+x2−1[x=1]{x^3+x^2-1}

Synthetic Division: Used this with the rational root test to solve third-order ordinary differential equations using the method of undetermined coefficients