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Area

• Calculation of surface area using integration

Cauchy's Residue Theorem

• Example 1
• Example 2

Chain rule

• Partial differentiation using the chain rule

Changing variables (integration)

• Solution of first order ODE by separation of variables

Completing the square

• Solution of first order ODE by separation of variables

Complex variable (function of)

• Taylor Series of a function of a complex variable
• Cauchy's Residue Theorem: Example 1
• Cauchy's Residue Theorem: Example 2
• Rouche's Theorem

Components (of a vector)

• Resolution of forces

Contour integration

• Cauchy's Residue Theorem: Example 1
• Cauchy's Residue Theorem: Example 2

Differentiation

• Differentiation of an integral
• Logarithmic differentiation

Differential Equations

• Solution of first order ODE using an integrating factor
• Solution of first order ODE by separation of variables
• Solution of a second order ordinary differential equation with constant coefficients
• Solution of Euler's Equation (a second order nonlinear ordinary differential equation)
• Introduction to the Frobenius method for solving a second order linear ordinary differential equation
• Solution of a second order linear ordinary differential equation using the Frobenius method

Definite integration

• Integration of sin x from 0 to π/2

Forces

• Resolution of forces

Frobenius method

• Introduction to the Frobenius method for solving a second order linear ordinary differential equation
• Solution of a second order linear ordinary differential equation using the Frobenius method

Generating functions

• Legendre polynomials and generating functions

Hyperbolic functions

• Integration by substitution using hyperbolic functions

Integrating factor

• Solution of first order ODE using an integrating factor

Integration over an area

• Calculation of surface area using integration

Lagrange multipliers

• Lagrange multipliers for optimisation with constraints

Legendre polynomials

• Legendre polynomials and generating functions

L'Hospital's rule

• Evaluation of limits using L'Hospital's rule

Limits

• Evaluation of limits using L'Hospital's rule

Line integral

• Line integral (given in terms of dx, dy, dz)
• Line integral (given in terms of dr)

Linear Differential Equation

• Solution of first order ODE using an integrating factor

Logarithms

• Logarithmic differentiation

Partial differentiation

• Partial differentiation using the chain rule
• Taylor Series of a function of two variables

Partial fractions

• Partial fractions

Regular singular point

• Introduction to the Frobenius method for solving a second order linear ordinary differential equation
• Solution of a second order linear ordinary differential equation using the Frobenius method

Repeated integral

• Calculation of surface area using integration

Residue

• Cauchy's Residue Theorem: Example 1
• Cauchy's Residue Theorem: Example 2

Rouche's Theorem

• Rouche's Theorem

Separation of variables

• Solution of first order ODE by separation of variables

Taylor Series

• Taylor Series of a function of two variables
• Taylor Series of a function of a complex variable

Trigonometric functions

• Integration of sin x from 0 to π/2
• Integration involving sin² x
• Integration involving sin³ x
• Integration involving tan³ x
• Integration involving products of sin x and cos x

Vectors

• Resolution of forces

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