∞ Calculus II

1st and 2nd order ODEs, calculus on curves, differential calculus in Rⁿ, vector functions, multiple integrals, vector fields, Fourier series and transforms. Every topic with step-by-step theory, worked examples and solved exercises.

📋 Formulario

Complete Theory

Worked Examples

Example 1Radius of convergence with two methods
Example 2Taylor expansion of exsin⁡xe^x\sin x
Example 3Fourier series of a square wave — convergence and Gibbs
Example 4RLC circuit with Laplace
Example 5Fourier series of f(x)=x2f(x)=x^2 and Parseval identity

Exercises with Solutions

Exercise 1Radius of convergenceMedium
📋 Problem to solve
Find the radius of convergence of ∑n=1∞nnn!xn\displaystyle\sum_{n=1}^\infty \frac{n^n}{n!} x^n.
Exercise 2Taylor seriesMedium
📋 Problem to solve
Expand f(x)=arctan⁡xf(x) = \arctan x in Maclaurin series up to 5th order.
Exercise 3FourierHard
📋 Problem to solve
Find the Fourier series of f(x)=∣x∣f(x)=|x| on [−π,π][-\pi,\pi] (period 2π2\pi).
Exercise 4LaplaceHard
📋 Problem to solve
Solve y′′+4y=sin⁡ty'' + 4y = \sin t, y(0)=0y(0)=0, y′(0)=1y'(0)=1 with Laplace.
Exercise 5Power seriesMedium
📋 Problem to solve
Compute ∑n=0∞(−1)n3n(2n)!\displaystyle\sum_{n=0}^\infty \frac{(-1)^n}{3^n (2n)!}.
Exercise 6FourierHard
📋 Problem to solve
Prove the identity ∑n=1∞1n2=π26\displaystyle\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6} using the Fourier series of f(x)=x2f(x)=x^2.
Exercise 7LaplaceMedium
📋 Problem to solve
Solve y′+3y=e−ty' + 3y = e^{-t}, y(0)=2y(0)=2 with Laplace.
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Recommended Books

Introductory
Calculus: Early Transcendentals
Stewart, Clegg, Watson
Buy on Amazon →
Intermediate
Multivariable Calculus
Stewart, Watson, Clegg
Buy on Amazon →

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