Tag Archives: line integral

Math2111: Chapter 3: Additional Material: Rectifiable parameterised curves

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 curves and their lengths in more detail. Let n\in\mathbb{N} be an arbitrary natural number which is fixed throughout this post. Note that in general one needs to distinguish between a parameterised curve, which is a continuous mapping \boldsymbol{c}:[a,b]\to\mathbb{R}^n, and the curve \mathcal{C}, which is the image of \boldsymbol{c} given by \{\boldsymbol{c}(t) \in \mathbb{R}^n: t\in [a,b]\}. Here we shall discuss parameterised curves. Hence, for instance, the parameterised curve \boldsymbol{c}(t)= \cos t \widehat{\boldsymbol{i}} + \sin t \widehat{\boldsymbol{j}} with 0 \le t \le 4\pi is a circle traversed twice and has therefore length 4\pi, whereas its image is just a circle which has length 2\pi. Continue reading

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Math2111: Chapter 3: Recommended reading: Line integrals and orientation

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 in what sense the scalar line integrals do not depend on the orientation of the curve.

Let us start with integrals \int_a^b f(t)\,\mathrm{d} t where f is a Riemann integrable function. Let a \textless b and let P_n=\{x_0,\ldots, x_n\} be such that a=x_0 \textless x_1 \textless x_2 \textless \cdots \textless x_{n-1} \textless x_n =b, let x_{i-1} \le t_i \le x_{i} for 1\le i \le n and \delta(P_n)=\max_{1\le i\le n} x_{i+1}-x_i. We can now form the Riemann sum

\displaystyle \mathcal{S}(P_n) = \sum_{i=1}^n f(t_i) (x_i-x_{i-1}).

By considering the limit \lim \mathcal{S}(P_n) where n\to\infty such that \delta(P_n) \to 0, we obtain the Riemann integral \int_a^b f(t)\,\mathrm{d} t. Continue reading

Math2111: Chapter 3: Line integrals. Section 2: Vector 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.

Work integral

Previously we considered integrals of a function f:\mathbb{R}^n\to\mathbb{R} (n=2,3) over a curve \mathcal{C}. We now turn to integrating vector fields \mathbb{F}:\mathbb{R}^n\to\mathbb{R}^n where n=2 or {}3. These integrals can be motivated by calculating the work done by a force on a particle along some curve \mathcal{C}.

In the simplest case, the work {}W done a force \mathbb{F} acting on an object which moves along a straight line is given by

\displaystyle W = \mathbb{F} \cdot \frac{\vec{d}}{\|\vec{d}\|} \|\vec{d}\|=\mathbb{F}\cdot\vec{d},

where the object is displaced in the direction of the vector \vec{d} with distance \|\vec{d}\|. Continue reading

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