Week 8: Preparation#

Key Concepts#

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

  • The Riemann Integral for Scalar Functions of \(n\) variables

  • The Riemann Integral for Vector Functions

  • The Change-of-Variables Theorem: Coordinate Change in \(\mathbb{R}^n\)

  • The Jacobian Determinant

  • Typical coordinates:

    • In \(\mathbb{R}^2\): Cartesian and Polar Coordinates

    • In \(\mathbb{R}^3\): Cartesian, Spherical, Cylindrical/Semi-Polar Coordinates

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:

  • Reading: The rest of chapter 6

  • Python demo08


Preparatory Exercises#

II: Antiderivative of Function of Multiple Variables?#

Let \(f:[0,2]\times[0,3]\to\Bbb R\) be given by \(f(x,y)=xy^2\). In this exercise we will be discussing whether \(f\) has an antiderivative. As we remember, an antiderivative is a differentiable function whose derivative is \(f\), and an antiderivative can be used for easy calculation of a definite integral of \(f\).

Question a#

If \(f\) has an antiderivative \(F\), what is then the domain of \(F\)?

Question b#

What is “the derivative” of such an antiderivative \(F\)? Will it be equal to \(f\)?

II: Image Set of a Parametrization#

Consider the parametrization

\[\begin{equation*} \pmb{r}(u,v) = \begin{bmatrix} u \\ 2uv \end{bmatrix}, \quad u \in [0,2], \quad v \in [0,1]. \end{equation*}\]

Question a#

Describe the parametrized region \(\operatorname{im}(\pmb{r})\).

III: Mathematical Parametrization of Earth#

Let us define a coordinate system such that the center of the Earth is placed at the origin, the \(x\)-axis points towards the prime (zero) meridian at the equator, and the \(z\)-axis points north. Let \(R\) be the Earth’s radius, and let \(d\) be the depth below the surface.

Note

Keep in mind that the prime meridian spans from the North Pole to the South Pole through the Greenwich Observatory at longitude \(0^\circ\).

Question a#

Provide a parameterization for the northern hemisphere from longitude \(0^\circ\) to longitude \(45^\circ\) and from a depth \(d\) below the surface to the surface at \(R\).

Question b#

How would you go about finding the volume of the region described in the previous question?