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Second Year+ — Formula Reference Sheet

Second Year University & Beyond · 7 formulas

Second Year University & Beyond

Second Year+

Multivariable calculus, ODEs, real analysis, abstract algebra, advanced statistics.

7 formulas · opens your print dialog

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Multivariable Calculus

Partial Derivative

\frac{\partial f}{\partial x} - d i f f er e n t ia t e f (x, y) w . r . t . x, treating y a sco n s t an t

Rate of change of a multivariable function with respect to one variable.

When to use

Optimisation of functions of multiple variables, gradient vectors.

Example

f(x,y) = x²y + y³ → ∂f/∂x = 2xy

Multivariable Calculus

Gradient Vector

\nabla f = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z})

Vector of all partial derivatives — points in the direction of steepest ascent.

When to use

Gradient descent in ML, directional derivatives, finding maxima/minima.

Example

f(x,y) = x²+y²: ∇f = (2x, 2y)

Multivariable Calculus

Multivariable Chain Rule

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

Chain rule extended to functions of multiple variables.

When to use

Composite functions of multiple parameters.

Example

z = f(x(t), y(t))

Differential Equations

Separable ODE

\frac{d y}{d x} = g (x) h (y) \to \int \frac{d y}{h ( y )} = \int g (x) d x

Solves ordinary differential equations by separating variables.

When to use

Population growth models, radioactive decay, Newton cooling law.

Example

dy/dx = ky → ∫dy/y = k∫dx → ln|y| = kx + C → y = Aeᵏˣ

Fourier Analysis

Fourier Series

f (x) = \frac{a _{0}}{2} + \sum [a_{n} cos (nπ \frac{x}{L}) + b_{n} sin (nπ \frac{x}{L})]

Represents any periodic function as an infinite sum of sinusoids.

When to use

Signal processing, PDEs, heat equation, wave equation.

Example

Square wave decomposed into sin harmonics: engineering acoustics

Complex Analysis

Cauchy's Integral Formula

f (z_{0}) = (\frac{1}{2 π} i) \oint \frac{f ( z )}{z - z _{0}} d z

Evaluates the value of a complex function inside a contour from its boundary values.

When to use

Complex integration, residue theorem applications.

Example

Used to evaluate real integrals that would be impossible by standard methods

Optimisation

Lagrange Multipliers

\nabla f = λ \nabla g and g (x, y) = c

Finds maxima/minima of f subject to a constraint g = c.

When to use

Constrained optimisation in economics, engineering design, data science.

Example

Maximise profit f(x,y) subject to budget constraint g(x,y) = c

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