Monthly Archives: April 2010

Math2111: Chapter 3: Line integrals. Section 1: Scalar line integrals

In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

We now discuss the concept of a scalar line integral. This is one of several ways in which integrals of functions of one variable can be generalized to functions of several variables. Here, instead of integrating over an interval {}[a,b] on the {}x axes, we now integrate over a curve \boldsymbol{c}:[a,b]\to\mathbb{R}^3. The integrand is then a function f:D\subseteq\mathbb{R}^3\to\mathbb{R} such that D\supseteq \{\boldsymbol{c}(t): t\in [a,b]\}. Continue reading

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Walsh functions I. Orthonormality and completeness

In this post I summarize some useful properties of Walsh functions. These functions were introduced by Joseph Walsh in

  1. J. L. Walsh, A closed set of normal orthogonal functions. Amer. J. Math., 45, 5-24, 1923.

Another paper where many ideas can be found is by Nathan Fine

  1. N. J. Fine, On the Walsh functions. Trans. Amer. Math. Soc., 65, 372-414, 1949.

In this exposition here we only concentrate on the simplest case of base b = 2 and dimension s = 1.

We write \mathbb{N} for the set of natural numbers 1,2,3, \ldots and \mathbb{N}_0 for the set of nonnegative integers 0,1,2,\ldots.

A table of contents for the posts on Walsh functions can be found here.

Definition of Walsh functions Continue reading

Math2111: Chapter 2: Vector fields and the operator ∇. Section 2: Divergence and Curl

In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

We discuss now derivatives of vector fields. To that end we introduce the del operator ∇. Throughout we assume that the vector field \boldsymbol{F}:\mathbb{R}^3\to\mathbb{R}^3 and the scalar field f:\mathbb{R}^3\to\mathbb{R} are \mathcal{C}^1, i.e. have at least one continuous partial derivative in each coordinate (similarly we write \mathcal{C}^2 if the functions are assumed to have continuous partial derivative of order at least two). Continue reading

Math2111: Chapter 2: Vector fields and the operator ∇. Section 1: Vector fields

In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

In this part we deal with vector fields. These are useful for describing fluid flow or force.

Imagine a fluid moving in a pipe in a steady flow. (In physical applications the vector field may also depend on time, but we do not consider such instances here.) Then at each point we can draw a velocity vector corresponding to the velocity of the fluid at this particular point to obtain the velocity field of the fluid. If we put a particle in the fluid, then, at each point, the direction of the velocity vector is tangent to the movement of the particle and the length of the vector corresponds to the speed of the particle. See
here for a picture on fluid flow or here for some animations on fluid flow. Similarly one can draw force fields, see for instance here and here. The force field of a magnetic dipole is shown here. Continue reading

Math2111: Chapter 1: Fourier series. Recommended reading: Motivation of formulae for Fourier series and a comparison to Taylor series

In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

We motivate the formulae for Fourier series. For simplicity we ignore here all questions concerning convergence, which are dealt with in Section 3 and the additional material.

Fourier series

As pointed out in the post on Section 2 at the beginning, in order to represent a function {}f we need an orthonormal bases with respect to the inner product

\displaystyle \langle f, g \rangle = \int_{-\pi}^\pi f(x) g(x) \, \mathrm{d} x.

Continue reading

Math2111: Chapter 1: Fourier series. Section 6: Heat equation

In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

In this part we discuss applications of Fourier series to solving a certain type of partial differential equation (pde). In more detail, we discuss the heat equation. The aim is to show how Fourier series naturally come up in the solution of this equation. Continue reading

Math2111: Chapter 1: Fourier series. Additional Material: L_2 convergence of Fourier series.

In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

In this post I present some ideas which shed light on the question why one can expect the Fourier series to converge to the function (under certain assumptions). Continue reading