Week 9: Preparation#

Key Concepts#

After reading, you should be able to explain the following key concepts:

  • Parametric representations of curves and surfaces in \(\mathbb{R}^n\)

  • Curve length

  • The normal vector of a surface

  • The line integral and the surface integral

  • The antiderivative problem in \(\mathbb{R}^n\)

  • Vector fields and gradient vector fields

  • Flux

Note

At DTU (but not much elsewhere), the line integral of a vector field along a curve is often called the tangential line integral.

This week, we will explore these key concepts in great detail. We expect you to have familiarized yourself with these topics before lectures.

Reading Material#

We recommend that you read the textbook. Watching YouTube videos on the week’s topics can be useful, but it should not replace proper preparation for the week’s program and is not recommended as a standalone approach.

Read and study the following:


Preparatory Exercises#

1: Parametric Representation and Curve Length of a Circle#

Let \(\mathcal{C}\) denote a circle in \(\mathbb{R}^2\) given by the equation

\[\begin{equation*} (x-1)^2 + y^2 = 4. \end{equation*}\]

Question a#

State the center and radius of \(\mathcal{C}\).

Question b#

Choose a parametric representation \(\pmb{r}(t)\) for \(\mathcal{C}\) with \(t \in [0, 2\pi]\).

Question c#

We know that its curve length is \(2\pi r\), so \(4 \pi\), as this is the well-known circumference formula for a circle. We here want to rediscover this value using the general formula for curve length. First, determine the Jacobian function, meaning the norm of \(\pmb{r}'(t)\), and calculate the curve length of \(\mathcal{C}\) using a fitting formula from the textbook.

2: Line Integral of Scalar Function#

Let \(f(x,y)=x^2+y^2\), and let \(\mathcal{C}\) be the same circle as in exercise 1: Parametric Representation and Curve Length of a Circle with the parametric representation

\[\begin{equation*} \pmb{r}(t)=\bigl(1+2\cos(t),\,2\sin(t)\bigr),\quad t\in[0,2\pi]. \end{equation*}\]

Question a#

Find the expression for \(f(\pmb{r}(t))\).

Question b#

Calculate the line integral

\[\begin{equation*} \int_{\mathcal{C}} f(x,y)\,\mathrm ds. \end{equation*}\]

3: Determination of a Gradient Field#

Consider the vector field \(\pmb{V}: \mathbb{R}^2 \to \mathbb{R}^2\), \(\pmb{V}(x,y)=(2xy,\,x^2)\).

Question a#

Can you guess a function \(f: \mathbb{R}^2 \to \mathbb{R}\) such that \(\nabla f = \pmb{V}\)?

Question b#

Maybe you cannot easily guess whether an \(f\) exists such that \(\nabla f = \pmb{V}\). So, first, find a way to investigate whether \(\pmb{V}\) is a gradient field. Then, if it is, find an antiderivative \(f(x,y)\), such that \(\nabla f = \pmb{V}\).