Week 8: Preparation

Week 8: Preparation#

Reading Material#

Key Concepts#

Long Day will cover:

  • Plane integrals: The Riemann integral for scalar functions with \(2\) variables

  • The change-of-variables theorem: coordinate change in \(\mathbb{R}^2\)

  • The Jacobian determinant and the Jacobian “function”

  • Polar, spherical, and cylindrical coordinates

Short Day will cover:

  • Volume integrals: The Riemann integral for scalar functions with \(3\) variables

  • The Riemann integral for scalar functions with \(n\) variables

  • The change-of-variables theorem: coordinate change in \(\mathbb{R}^n\)

Preparatory Exercises#

I: 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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