Week 8: Exercises#
Exercises – Long Day#
Note
The expression \(|\det{\pmb{J}_{\pmb{r}}(\pmb{u})}|\) is in mathematical literature often referred to as “the Jacobian”. This phrase is also often used for the Jacobian matrix, though. To avoid confusion, we will at DTU refer to \(|\det{\pmb{J}_{\pmb{r}}(\pmb{u})}|\) as “the Jacobian function”, although this is not a phrase you will see in literature outside of DTU. Alternatively, we might simply refer to it in full: “the absolute value of the determinant of the Jacobian matrix”.
1: Plane Integrals over Rectangles. By Hand#
Question a#
Consider the region \(B=\left\lbrace (x,y) \bigm| 0\leq x\leq 2 \wedge -1\leq y\leq 0\right\rbrace\) in \(\mathbb{R}^2\). Calculate the plane integral
using the formula for double integrals over (axis-parallel) rectangles.
Question b#
Let us calculate the plane integral from above one more time, but now in a manner that at first glance may appear more complicated. Use the change-of-variables theorem for integrals over \(\mathbb{R}^2\).
Hint
First, find a parametrization of the region. Notice that \(B\) is an axis-parallel rectangle in the \((x,y)\) plane.
Hint
A possible parametrization is simply \(\pmb{r}(u,v)=(2u,v-1)\), where \((u,v) \in [0,1]^2\). An alternative is \(\pmb{r}(u,v)=(2u,-v)\), where \((u,v) \in [0,1]^2\). Have a try with both.
Hint
Find the Jacobian function (the absolute value of the Jacobian determinant). Then express the integrand \(x^2y^2+x\) by \(u\) and \(v\).
Hint
Remember the link \((x,y) = \pmb{r}(u,v)\).
Answer
Question c#
Calculate the plane integral
Hint
Notice that \(B\) once again is an axis-parallel rectangle in the \((x,y)\) plane.
Hint
Hint
With respect to the innermost integral, use integration by substitution where your inner function could be
and your outer function could be
Answer
2: Polar Coordinates. By Hand#
A function \(f:\mathbb{R}^2 \to \mathbb{R}\) is given by
For a given point \(\pmb{x}=(x,y)\) in the plane, let \(r = \Vert \pmb{x} \Vert\) denote the distance from the point to the origin \((0,0)\). Also, let \(\theta\) denote the angle between the \(x\)-axis and the position vector to the point - for the sign of the angle \(\theta\), we define the positive angular orientation as counterclockwise. A set of points \(B\) contains all points that fulfill (in polar coordinates),
where \(a\) is an arbitrary positive real number.
Question a#
Make a sketch of \(B\), and determine the area of \(B\), first using integration and then purely from elementary geometric considerations.
Hint
You may use this parametric representation of \(B\): \((x,y)=\pmb{p}(u,v)=(u\cos(v),u\sin (v))\) for \(-\frac{\pi}{4} \leq v \leq \frac{\pi}{2}\) and \(0\leq u \leq a\). If you wish, you may of course rename \(u\) to \(r\) and \(v\) to \(\theta\) (which are often-used symbols when dealing with polar coordinates).
Answer
The area is \(\frac{3}{8} a^2\pi\).
Question b#
Determine the plane integral \(\int_B f(x,y) \;\mathrm{d}\pmb{x}\).
Hint
By hand: For finding antiderivatives the following trigonometric formula for doubled angles may be of use:
Answer
3: The Volume of a Parallelotope#
A parallelotope \(P\) in \(\mathbb{R}^n\) “spanned by” the vectors \(\pmb{a}_1, \pmb{a}_2, \dots, \pmb{a}_n\) is defined by:
where \(A = [\pmb{a}_1 | \pmb{a}_2 | \cdots | \pmb{a}_n]\) is the \(n \times n\) matrix whose \(i\)’th column is \(\pmb{a}_i\). This set of points can in short-hand notation be written as \(P=A([0,1]^n)\).
It can be shown with tools solely from Mathematics 1a (in particular the characterization of the determinant) that the \(n\)-dimensional volume of \(P\) is:
(For the interested student, such a proof can be found here https://textbooks.math.gatech.edu/ila/determinants-volumes.html)
In \(\mathbb{R}^2\), a parallelotope is the well-known pallelogram, and \(\mathrm{vol}_n(P)\) is then the area of \(P\), while it in \(\mathbb{R}^3\) becomes a parallelepiped with a volume.
Question a#
Show that \(\mathrm{vol}_n(P) = |\mathrm{det}(A)|\) using the change-of-variables theorem for integrals over \(\mathbb{R}^n\).
Hint
Since \(P=A([0,1]^n)\), we will choose \(\Gamma = [0,1]^n\). What is the expression of the corresponding parametrization?
Hint
\(\pmb{r}(\pmb{u}) = A \pmb{u}\), where \(\pmb{u} \in \Gamma = [0,1]^n\). What is the corresponding Jacobian matrix?
Answer
The Jacobian determinant is \(\mathrm{det}(A)\). For computation of \(\mathrm{vol}_n(P)\), we need to use \(f(\pmb{x}) = 1\). The full integrand is thus the constant \(|\mathrm{det}(A)|\), which is to be integrated over \(\Gamma = [0,1]^n\). This just becomes \(|\mathrm{det}(A)|\). The integral we have calculated is by definition equal to \(\mathrm{vol}_n(P)\).
In the rest of this exercise we want to investigate the proposition \(\mathrm{vol}_n(P) = |\mathrm{det}(A)|\) without use of integration techniques.
Question b#
Let \(n=2\). Choose two linearly independent vectors \(\pmb{a}_1, \pmb{a}_2\) in \(\mathbb{R}^2\). It might be smart to choose \(\pmb{a}_1 \in \mathrm{span}(\pmb{e}_1)\). Calculate (using elementary geometric considerations) the area of the parallelogram “spanned by” the two vectors. Also calculate \(|\mathrm{det}(A)|\) and compare the results.
Hint
You may need to do an orthogonal projection of \(\pmb{a}_2\) on the orthogonal complement to \(\pmb{a}_1\).
Question c#
Let \(n=2\), and now let \(\pmb{a}_1, \pmb{a}_2\) be arbitrary but linearly independent vectors in \(\mathbb{R}^2\). Can you prove the formula \(\mathrm{area}(P) = |\mathrm{det}(A)|\), where \(P\) is the parallelogram “spanned by” the two vectors? You may assume (why?) that \(\pmb{a}_1 \in \mathrm{span}(\pmb{e}_1)\), if this helps in your argumentation.
Hint
You might again need to do the orthogonal projection of \(\pmb{a}_2\) on the orthogonal complement to \(\pmb{a}_1\).
Answer
Below is given a proof without use of trigonometric identities:
As rotation does not change the area of a region, then by rotating the parallelogram we can assume that \(\pmb{a}_1 \in \mathrm{span}(\pmb{e}_1)\), meaning that \(A\) is an upper triangular matrix \(A = [a_{i,j}]\) with \(a_{1,2}=0\). As \(A\) is an upper triangular matrix, its determinant is the product of its diagonal elements, so \(\mathrm{det}(A) = a_{1,1} a_{2,2}\). Hence, \(|\mathrm{det}(A)| = |a_{1,1} a_{2,2}|\). We now just need to prove that this i also equal to the area.
The area of the parallelogram is given as the length of \(\pmb{a}_1\) (meaning \(|a_{1,1}|\)) multiplied by “the height”, which is the length of the orthogonal projection of \(\pmb{a}_2\) on \(\mathrm{span}(\pmb{e}_2)\) (so, \(|\langle \pmb{a}_2, \pmb{e}_2 \rangle| = |a_{2,2}|\)). Hence, the area is: \(|a_{1,1}| |a_{2,2}| = |a_{1,1} a_{2,2}|\).
Question d#
Let \(n=3\). Choose three linearly independent vectors \(\pmb{a}_1, \pmb{a}_2, \pmb{a}_3\) in \(\mathbb{R}^3\). It can be smart to choose \(\pmb{a}_1, \pmb{a}_2 \in \mathrm{span}(\pmb{e}_1, \pmb{e}_2)\). Calculate (using elementary geometric considerations) the volume of the parallelepiped “spanned by” the three vectors. Also calculate \(|\mathrm{det}(A)|\) and compare the two results.
Hint
You may need to do an orthogonal projection of \(\pmb{a}_2\) on the orthogonal complement to \(\pmb{a}_1\).
Question e (Extra, can Wait Until After the Exercises of the Day)#
Let \(n=3\), and now let \(\pmb{a}_1, \pmb{a}_2, \pmb{a}_3\) be arbitrary but linearly independent vectors in \(\mathbb{R}^3\). Can you prove the formula \(\mathrm{areal}(P) = |\mathrm{det}(A)|\), where \(P\) is the parallelepiped “spanned by” the three vectors? You may assume (why?) that \(\pmb{a}_1, \pmb{a}_2 \in \mathrm{span}(\pmb{e}_1, \pmb{e}_2)\), if that helps your argumentation.
4: Plane Integral with Parametrization I. By Hand#
In the \((x,y)\) plane we are given the point \(P_0=(1,2)\) and the set of points
Question a#
Make a preliminary sketch of \(C\) and provide a parameterization \(\pmb{r}(u,v)\) for \(C\) with appropriate intervals for \(u\) and \(v\), i.e., specify \(\Gamma\) such that \(\pmb{r}(\Gamma) = C\). Justify that the chosen parameterization is injective (if the chosen parameterization is not injective, you must find a new one).
Question b#
Determine the two parameter values \(u_0\) and \(v_0\) such that \(\pmb{r}(u_0,v_0)=P_0\). Make an illustration of \(C\) (both a sketch by hand and a plot in Sympy are fine) where you from \(P_0\) draw the tangent vectors \(\pmb{r}'_u(u_0,v_0)\) and \(\pmb{r}'_v(u_0,v_0)\). Determine the area of the parallelogram that is spanned by these tangentvectors, see according to Exercise 3: The Volume of a Parallelotope.
Question c#
Determine the Jacobian determinen that corresponds to \(\pmb{r}(u,v)\), and argue that the two column vectors that constitute the Jacobian matrix are linearly independent for all \((u,v) \in \Gamma\). Calculate the Jacobian determinant at the point \((u_0,v_0)\).
Question d#
Calculate the plane integral
using the change-of-variables theorem for integrals over \(\mathbb{R}^2\). You must argue that changing variables is a usable method for this case. Check your result with the theorem on axis-parallel regions.
Answer
The parametrization can for example be \(\pmb{r}(u,v)=(\tfrac{1}{2}vu^2,u)\), where \(u\in\left[ \tfrac 32,\tfrac 52\right]\) and \(v\in\left[ 0,1\right]\). The corresponding Jacobian determinant is then \(-\frac{1}{2}u^2\).
5: The Plane Integral with Parametrization II#
We want to determine the plane integral
given by the parametrization
Follow the below steps.
Question a#
Describe the region \(B\) using inequalities (such as \(x+5y\ge 7\)). Then sketch \(B\).
Hint
Fix \(u\) and you will see that \(\pmb{r}(u,v)\) runs through a line as \(v\) is varied.
Answer
The region \(B\) can easily be sketched based on the following inequalities:
Question b#
Determine the Jacobian determinant for the parametrization \(\pmb{r}(u,v)\). Is the Jacobian determinant different from zero on the interior of the parameter domain (this is a requirement for using the change-of-variables theorem)?
Answer
\(\mathrm{det}(\pmb{J}_{\pmb{r}}(u,v)) = 1-u > 0\). The Jacobian determinant is positive for \(u,v \in ]0,1[\), so on the interior of the parameter domain \(\Gamma^\circ\), where \(\Gamma = [0,1]^2\).
Question c#
Now determine the wanted integral.
Hint
You must substitute the first and second coordinates of the parametrization into the function \(2xy,\) in place of \(x\) and \(y\), respectively, and then multiply by (the absolute value of) the Jacobian determinant. Then you must do the integration, first with respect to \(u\), then to \(v\).
Hint
You must integrate
Hint
An antiderivative with respect to \(u\) is:
When you substitute in the \(u\) limits, you get \(\displaystyle{\frac 16 v}\) which you now must integrate with respect to \(v\).
Answer
6: A Triple Integral#
Calculate the triple integral
Hint
We integrate over an axis-parallel box!
Answer
\(\displaystyle{\frac 94 \ln(2).}\)
7: Partial Integration and Integration by Substitution in Two Variables#
Question a#
Determine \(\displaystyle{\int_0^{\frac{\pi}{2}}\left(\int_0^{\frac{\pi}{2}} u\cos(u+v)\mathrm{d}u\right)\mathrm{d}v.}\)
Hint
Find using partial integration with respect to \(u\) an antiderivative \(F(u)\) of the function
Then
will be a function of \(v\) that you now has to find an antiderivative \(G(v)\) of. Then substitute in the limits of \(v\).
Hint
\(G(v)=\frac{\pi}{2}\cos(v)-\sin(v)-\cos(v)\).
Answer
\(\displaystyle{\frac{\pi}{2}-2}\).
Question b#
Determine \(\displaystyle{\int_0^1\left(\int_0^1 \frac{v}{(uv+1)^2}\mathrm{d}u\right)\mathrm{d}v.}\)
Hint
Find using integration by substitution with respect to \(u\) an antiderivative \(F(u)\) of the function
Then
will be a function of \(v\) that you now have to find an antiderivative \(G(v)\) of. Then substitute in the limits of \(v\).
Hint
\(\displaystyle{G(v)=1-\frac{1}{v+1}}\).
Answer
\(1-\ln(2)\).
Exercises – Short Day#
1: Parametrized Spatial Region. By Hand.#
A region \(B\) in \((x,y,z)\) space is given by the parametric representation
Question a#
In \(B\) we are given the point
Find \(\pmb{x}_0\). When placed at \(\pmb{x}_0\), the tangent vectors \(\pmb{r}_u'(1,1,1),\pmb{r}_v'(1,1,1)\) and \(\pmb{r}_w'(1,1,1)\) span a parallelepiped \(P\), see Exercise 3: The Volume of a Parallelotope. Determine the volume of this parallelepiped. It would be good training to illustrate this with Sympy.
Hint
The volume is found as the absolute value of the determinant af the matrix whose columns are the spanning vectors.
Answer
\(\pmb{x}_0=(-1/2, -1, 1)\). The volume of \(P\) is \(3\).
Question b#
Determine the absolute value of the Jacobian determinant corresponding to \(\pmb{r}\). Evaluate it at \(\pmb{x}_0\).
Hint
Remember that the Jacobian determinant is the determinanten of the Jacobian matrix. How is this matrix related to what we found in the previous question?
Answer
\(|\pmb{J}_{\pmb{r}}(u,v,w)|=u^2+2v^2\). Altså er \(|\pmb{J}_{\pmb{r}}(1,1,1)|=1^2+2\cdot 1^2 = 3\).
Question c#
Calculate the volume of \(B\).
Hint
The volume of \(B\) can be calculated as the integral over the function 1 - and don’t forget the absolute value of the Jacobian determinant as the integrand.
Answer
\(32\).
2: Mass Distributions in the \((x,y)\) Plane#
Consider the sets of points in \(\mathbb{R}^2\) given by:
and consider (again)
We will think of \(f(x,y)\) as a function that expresses the mass density at the point \((x,y)\) (so, with units such as \(\mathrm{kg/m^2}\)).
Question a#
Assume that the mass density is constant, \(f(x,y)=1\) for \((x,y)\in B\). Determine the mass and centre of mass of \(B\).
Hint
The mass is \(M = \int_B 1 \mathrm{d}(x,y)\). Find the centre-of-mass formula in the textbook in the section about integration of vector functions.
Answer
The mass is \(\frac{15}{4}\) and the centre of mass is \((x,y)=(\frac{124}{75},\frac{254}{105})\).
Question b#
Assume that the mass density is \(f(x,y)=x^2\) for \((x,y)\in B\). Determine the mass and the centre of mass of \(B\).
Answer
The mass is \(\frac{21}{2}\) and the centre of mass is \((x,y)=(\frac{254}{147},\frac{73}{27})\).
Question c#
Assume that the mass density is constant \(f(x,y)=1\) for \((x,y)\in C\). Determine the mass and the centre of mass of \(C\).
Answer
The mass is \(\frac{49}{24}\) and the centre of mass is \((x,y)=(\frac{4323}{3920},\frac{102}{49})\).
Question d#
Assume that the mass density is \(f(x,y)=x^2\) for \((x,y)\in C\). Determine the mass and the centre of mass of \(C\).
Answer
The mass is \(\frac{37969}{10752}\) and the centre of mass is \((x,y)=(\frac{6767047}{3645024},\frac{84014}{37969})\).
3: Spherical Regions in 3D Space#
Consider the spatial region \(\pmb{r}(\Gamma)\) given by
where \(\Gamma = [a,b] \times [c,d] \times [e,f] \subset [0, \infty[ \times [0,\pi] \times [0,2\pi]\). Meaning, we are dealing with the following parameter values: \(u\in [a,b],v\in [c,d],w\in [e,f]\).
Question a#
What do the parameters represent?
Question b#
Let \(A\) be the region that is determined by the choice:
and let \(B\) be the region determined by the choice:
Describe in words each of the regions \(A\), \(B\) and \(A\cap B\), and calculate their volumes.
Answer
\(\mathrm{vol} (A)=\frac{13\pi(\sqrt 2-1)}{4}\).
\(\mathrm{vol} (B)=\frac{14\pi\sqrt 2}{3}\).
\(\mathrm{vol} (A\cap B)=\frac{19\pi(\sqrt 2-1)}{24}\).
By the way, \(\mathrm{vol}(A\cup B)\) can be calculated as \(\mathrm{vol}(A) + \mathrm{vol}(B) - \mathrm{vol}(A\cap B)\).
Question c#
Let \(\symbols x=(x_1,x_2,x_3)\). Calculate all of the integrals
Answer
4: An Indefinite Integral in the Plane#
Let \(B\) be the unit square \([0,1]^2\). We will investigate the indefinite plane integral:
The integrand \(f(x_1,x_2)=\frac{1}{x_2-x_1-1}\) is not Riemann integrable over \(B\), since \(f\) is not defined at the point \((x_1,x_2)=(0,1)\). We wish to find out whether it is still possible to give the integral a value by considering limits.
Question a#
Find those points in the \((x,y)\) plane where \(f(x_1,x_2)\) is not defined. Find the range of \(f\) as a function on \(B \setminus \{(0,1)\}\).
Hint
Is \(f\) positive or negative on \(B\)?
Question b#
Let \(B_a = [a,1] \times [0,1]\) for a fixed \(a \in [0,1]\). Make a sketch of \(B_a\) and create a parametrization of \(B_a\). Determine the Jacobian determinant of the parametrization.
Question c#
Calculate the Riemann integral
for every \(a \in ]0,1]\).
Question d#
Calculate the limit of \(I_a\) for \(a \to 0\).
Answer
\(I = -2 \ln(2)\)
Question e#
Let \(B_b = [0,1] \times [0,b]\). Define \(I_b := \int_{B_b} \frac{1}{x_2-x_1-1} \mathrm{d}\pmb{x} \). Find \(\lim_{b \to 1} I_b\). Compare with the above.