Week 8: Preparation

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 \(F\)?

Answer

Since \(f(x,y)=xy^2\) is a function of two variables, \(F\) must also be a function of two variables, \(F(x,y)\). The domain of \(F\) must be the same as that of \(f\), so \([0,2]\times[0,3]\).

Question b#

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

Answer

“The derivative” of \(F\) would be its gradient, \(\nabla F(x,y)\). The gradient is a vector in \(\Bbb R^2\) consisting of the two partial derivatives, so it is not equal to \(f(x,y)\). The conclusion seems to be that the concept of antiderivatives cannot be directly generalized to scalar functions of multiple variables.

You can consider whether a function exists that can be used to calculate the definite integral of \(f\), which we would write as \(\int_{[0,2]\times[0,3]} f(x,y)\,\mathrm d(x,y)\), in an easy manner like in the case of one variable with \(F(b)-F(a)\).

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 \(\operatorname{im}(\pmb{r})\).

Hint

Keep \(u\) fixed. Determine all possible values of \(\pmb{r}(u,v)\) as \(v\) varies over its domain.

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 \(R\).

Hint

Consider how the \(y\)-axis is positioned.

Hint

Use spherical coordinates with a radius varying from \(R-d\) to \(R\).

Question b#

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

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