Multiple Integrals and Vector Fields for 5 Days

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This is a set of problems with which you can take exercise on multiple integrals and integrals of vector fields.

Day 1.

In problem 1-5, evaluate the integrals.

1. \[\int_1^2 \int_{-1}^1 \frac{x}{y^2} \,dx\,dy.\]
2. \[\int_0^{\ln 2} \int_0 ^{\pi/2} e^x\,\cos y \,dy\,dx.\]
3. \[\int_0^2 \int_{y/2}^1 e^{x^2} \,dx\,dy .\]
4. \[\int_0^\pi \int_0^\pi \int_0^\pi \cos(x+y+z) dx\,dy\,dz.\]
5. \[\int_{-1}^1 \int_0^{\sqrt{1-y^2}}\int_0^x (x^2 + y^2) dz\,dx\,dy.\]

The following problems 6-10 are related to the definition of multiple integrals, in the section 15.1.

6. Give an example of a continuous function \(f\) defined on the interior of \(D = [0,\,2] \times [0,\,1]\) for which both \[\int_0^1 \int_0^2 f(x,\,y) \,dx\,dy\] and \[\int_0^2 \int_0^1 f(x,\,y) \,dy\,dx\] exist, while two integrals do not coincide. (See $15.1 Exercises 35.)
7. Express the following integral by limits of summations. \[\int_0^1 \int_0^2 (x+2y-3) dx\,dy.\]
8. Express the following limit by a double integral and find the value. \[\lim_{m\rightarrow\infty}\lim_{n\rightarrow\infty}\sum_{j=1}^n \sum_{i=1}^m \left( 4- \frac{2i}{m} - \frac{j}{n}\right) \frac{2}{mn}.\]
9. Let \(R = [a,\,b] \times [c,\,d]\) be a rectangular region with positive area and \(f\) be a continuous function defined on \(R.\) Show that \[\iint_R f \,dA = \lim_{n\to\infty} \sum_{i=1}^n \sum_{j=1}^n f\left( a+ \frac{b-a}{n} i ,\, c+ \frac{d-c}{n} j\right) \frac{(b-a)(d-c)}{n^2}.\]
10. Determine whether the function \(f\) defined by \[f(x,\,y) = \begin{cases} x+y &\quad \text{if}\,\, (x,\,y) \in \mathbb{Q} \times \mathbb{Q} \\[5pt] 0 &\quad \text{if}\,\, (x,\,y) \notin \mathbb{Q} \times \mathbb{Q} \end{cases}\] is integrable on \(R=[0,\,2]\times [0,\,2]\) or not, and prove it.

Day 2.

1. Let \(R = [a,\,b] \times [c,\,d]\) be a rectangular region with positive area. Suppose that \(f\) and \(g\) are continuous real functions defined on \([a,\,b]\) and \([c,\,d]\) respectively. Prove that \[\iint_R f(x) g(y) dA = \left(\int_a^b f(x)dx \right) \left( \int_c^d g(y) dy \right).\]
2. Find the volume of the solid region bounded above by the paraboloid \(z=9-x^2 -y^2\) and below by the unit disk in the \(xy\)-plane with center \((0,\,0)\).
3. Let \(D\) be the "ice cream cone" region cut from the solid sphere \(\rho \le 1\) by the cone \(\phi = \pi /3 .\) Find the volume of \(D.\)

The following problems 4-8 are related to the moments and centers of mass, in the section 15.6.

4. A solid of constant density \(\delta = 1\) occupies the region \(D\) given in the previous problem. Find the solid's moment of inertia about the \(z\)-axis.
5. Find the center of mass of a thin plate of density \(\delta = 3\) bounded by the lines \(x=0,\) \(y=x\) and the parabola \(y=2-x^2\) in the first quadrant.
6. Find the moments of inertia about the coordinate axes of a thin rectangular plate of constant density \(\delta\) bounded by the lines \(x=3\) and \(y=3\) in the first quadrant.
7. Find the centroid of the region in the first quadrant bounded by the \(x\)-axis, the parabola \(y=2x^2,\) and the line \(x+y=4.\)
8. Find the moments of interia of the rectangular solid bounded by three coordinate planes and planes \(x=a,\) \(y=b,\) \(z=c\) where \(a,\) \(b,\) \(c\) are positive numbers, with respect to its edge by calculating \(I_x,\) \(I_y\) and \(I_z.\)
9. Let \(A\) and \(B\) be open subsets of \(\mathbb{R}^2.\) Show that both \(A\cup B\) and \(A\cap B\) are open. (See $14.1.)
10. Give an example of the sets \(A\) and \(B\) that satisfies: (i) Both \(A\) and \(B\) are simply connected subsets of \(\mathbb{R}^2 ;\) (ii) \(A \cap B \ne \varnothing ;\) (iii) \(A\cup B\) is not simply connected. (See $16.3.)

Day 3.

In problem 1-4, find a potential function \(f\) for the field \(\mathbf{F}.\) (See $16.3.)

1. \(\mathbf{F} (x,\,y,\,z) = 2x\,\mathbf{i} + 3y \,\mathbf{j} + 4z \,\mathbf{k}.\)
2. \(\mathbf{F} (x,\,y,\,z) = (y+z) \,\mathbf{i} + (z+x) \,\mathbf{j} + (x+y) \,\mathbf{k}.\)
3. \(\mathbf{F} (x,\,y,\,z) = e^{y+2z}( \mathbf{i} + x\,\mathbf{j} + 2x \,\mathbf{k}).\)
4. \(\mathbf{F} (x,\,y,\,z) = (y \sin z)\mathbf{i} + (x\sin z)\mathbf{j} + (xy\cos z)\mathbf{k}.\)
5. Show that \[\mathbf{F} = (e^x \sin yz - y\sin xy) \mathbf{i} + (ze^x \cos yz - x\sin xy) \mathbf{j} + (ye^x \cos yz)\mathbf{k}\] is conservative on \(\mathbb{R}^3,\) and find a potential function \(f\) for \(\mathbf{F}.\) (See $16.3.)

The problems 6-10 are asking the line integrals of scalar fields. (See $16.1.)

6. Find the line integral of \(f(x,\,y) = ye^{x^2}\) along the curve \(\mathbf{r}(t) = 4t\,\mathbf{i} - 3t\,\mathbf{j},\) \(-1 \le t \le 2.\)
7. Evaluate\[\int_C (x+y)ds\]where \(C\) is the straight-line segment \(x=t,\) \(y=(1-t),\) \(z=0,\) from \((0,\,1,\,0)\) to \((1,\,0,\,0).\)
8. Evaluate\[\int_C (xy+y+z)ds\]along the curve \(\mathbf{r}(t) = 2t\,\mathbf{i} + t\,\mathbf{j} + (2-2t)\mathbf{k},\) \(0\le t\le 1.\)
9. Find the line integral of \(f(x,\,y,\,z) = x+y+z\) over the straight-line segment from \((1,\,2,\,3)\) to \((0,\,-1,\,1).\)
10. Find the line integral of \(f(x,\,y,\,z) = 2xy + \sqrt{z}\) over the helix \(\mathbf{r}(t) = \cos t \,\mathbf{i} + \sin t \,\mathbf{j} + t\,\mathbf{k},\) \(0\le t\le \pi .\)

Day 4.

1. Let \(C\) be the curve defined by \(\mathbf{r}(t) = t^2 \,\mathbf{i} + t\,\mathbf{j} + \sqrt{t}\,\mathbf{k},\) \(0\le t\le 1,\) and \(\mathbf{F}\) be the vector field defined by \(\mathbf{F}(x,\,y,\,z) = z\,\mathbf{i} + xy\,\mathbf{j} - y^2 \,\mathbf{k}.\) Find the line integral of \(\mathbf{F}\) along the \(C\) in the direction of increasing \(t.\)
2. Evaluate\[\int_C (x-y)dx\]where \(C\) is the curve defined by \(x=t,\) \(y=2t+1,\) \(0\le t\le 3.\)
3. Evaluate\[\int_C (x^2 + y^2)dy\]where \(C\) is the polygonal path consists of two straight-line segments, connecting \((0,\,0),\) \((3,\,0)\) and \((3,\,3).\)
4. Let \(C\) be the curve given by \(\mathbf{r}(t) = t\,\mathbf{i} + t^2\,\mathbf{j} + \mathbf{k},\) \(0\le t\le 1.\) Find the integral \[\int_C \,x^2\,dx + xy\,dy + dz.\]
5. Find the work done by \(\mathbf{F} = xy\,\mathbf{i} + y\,\mathbf{j} - yz\,\mathbf{k}\) over the curve \(\mathbf{r}(t) = t\,\mathbf{i} + t^2 \,\mathbf{j} + t\,\mathbf{k},\) \(0\le t\le 1,\) in the direction of increasing \(t.\)
6. Find the flow along the curve \(\mathbf{r}(t) = t\,\mathbf{i} + t^2\,\mathbf{j} + \mathbf{k},\) \(0\le t\le 2\) in the direction of increasing \(t,\) if the velocity field of fluid is given as \(\mathbf{F} = -4xy\,\mathbf{i} + 8y \,\mathbf{j} +2\,\mathbf{k}.\)
7. Evaluate the integral\[\int_{(0,\,0,\,0)}^{(1,\,2,\,3)} 2xy\,dx + (x^2 - z^2)dy - 2yz\,dz.\]
8. Find the condition for constants \(a,\) \(b\) and \(c\) for which the differential form \[(ay^2 + 2czx)dx + y(bx+cz)dy + (ay^2 + cx^2)dz\] is exact.
9. Find the flux of the field \(\mathbf{F}(x,\,y) = 2e^{xy} \mathbf{i} + y^3 \mathbf{j}\) outward across the square with vertices \((1,\,1),\) \((1,\,-1),\) \((-1,\,-1)\) and \((-1,\,1).\)
10. Let \(D\) be a simply connected domain and \(\mathbf{F}\) be a vector field defined on \(D.\) Show that \(\mathbf{F}\) is conservative on \(D\) if and only if \[\int_C \mathbf{F} \cdot d\mathbf{r} = 0\] for any closed smooth simple curve lying in \(D.\)

Day 5.

1. Let \(R\) be a planar region surrounded by a smooth simple closed curve \(C,\) provided that counterclockwise orientation is given. Show that the arc of \(R\) is \[\frac{1}{2}\oint_C x\,dy - y\,dx.\] (See $16.4.)
2. Find the area of the surface cut from the paraboloid \(x^2 + y^2 - z=0\) by the plane \(z=2.\) (See $16.5.)
3. Find the surface area of a sphere with radius \(R\) by using spherical coordinates and a surface integral. (See $16.5.)
4. Integrate \(G(x,\,y,\,z) = xyz\) over the surface of the cube cut from the first octant by the plane \(x=1,\) \(y=1\) and \(z=1.\) (See $16.6.)
5. Find the flux of \(\mathbf{F} = yz\,\mathbf{j} - z^2 \,\mathbf{k}\) outward the surface \(S\) cut from the cylinder \(y^2 + z^2 = 1,\) \(z\ge 0,\) by the planes \(x=0\) and \(x=1.\) (See $16.6.)
6. Use the Stokes' Theorem to find the circulation of the field \(\mathbf{F} = (x^2 - y)\mathbf{i} + 4z\,\mathbf{j} + x^2\,\mathbf{k}\) around the curve \(C\) in which the plane \(z=2\) meets the cone \(z=\sqrt{x^2 + y^2},\) counterclockwise as viewed from above. (See $16.7.)
7. Use the Divergence Theorem to find the flux of \(\mathbf{F} = xy\,\mathbf{i} + yz\,\mathbf{j} + xz\,\mathbf{k}\) outward through the surface of the cube cut from the first octant by the planes \(x=1,\) \(y=1\) and \(z=1.\) (See $16.8.)
8. Let \(S\) be the surface of the portion of the solid sphere \(x^2 + y^2 + z^2 \le a^2\) that lies in the first octant and let \[f(x,\,y,\,z) = \ln\sqrt{x^2 + y^2 +z^2}.\] Calculate \[\iint_S \nabla f \cdot \mathbf{n}\,d\sigma.\] (It is recommended to try to calculate the integral in two different ways; the one is a direct calculation, and the other is by using the Divergence Theorem. See $16.8.)
9. Let \(\mathbf{F} = M\,\mathbf{i} + N\,\mathbf{j} + P\,\mathbf{k}\) be a vector field with continuous second partial derivatives. Show that\[\operatorname{div}(\operatorname{curl} \mathbf{F} ) = 0.\] (See $16.8.)
10. Deduce the Green's Theorem from the Stokes' Theorem. (See $16.4 and $16.7.)

Additional Problems.

1. Let \(a\) and \(b\) be constants with \(a > b > 0.\) Find the net ourward flux of the field \[\mathbf{F} = \frac{x\,\mathbf{i} + y\,\mathbf{j} + z\,\mathbf{k}}{\rho^3} ,\,\, \rho=\sqrt{x^2+ y^2 + z^2} \] across the boundary of the region \(b \le \rho \le a.\)
2. The electric field created by a point charge \(q\) located at the origin is \[\mathbf{E}(x,\,y,\,z) = \frac{q}{4\pi\epsilon_0} \mathbf{F}(x,\,y,\,z),\] where \((x,\,y,\,z)\) is the location of the point, \(\mathbf{F}\) is given in the previous problem and \(\epsilon_0\) is a physical constant. Show that the outward flux of \(\mathbf{E}\) across any closed surface \(S\) that encloses the origin and to which the Divergence Theorem applies is \(q/\epsilon_0 ,\) by using the result of the previous problem.
3. Suppose that \(f\) and \(g\) are scalar functions with continuous first- and second-order partial derivatives throughout a region \(D\) that is bounded by a closed piecewise smooth surface \(S.\) Show that \[\iint_S f \nabla g \cdot \mathbf{n} \,d\sigma = \iiint_D ( f \nabla ^2 g + \nabla f \cdot \nabla g) dV .\tag{*}\] This equation is called Green's first formula.
4. Interchange \(f\) and \(g\) in equation (*) of the previous problem to obtain a similar formula. Then subtract the formula from (*) to show that \[\iint_S (f\nabla g - g\nabla f) \cdot \mathbf{n} \,d\sigma = \iiint_D ( f\nabla^2 g - g\nabla^2 f) dV. \] This equation is called Green's second formula.
5. If \(\mathbf{E}(t,\,x,\,y,\,z)\) and \(\mathbf{B}(t,\,x,\,y,\,z)\) represent the electric and magnetic fields at point \((x,\,y,\,z)\) at time \(t,\) a basic principle of electromagnetic theory says that \(\nabla \times \mathbf{E} = - \partial\mathbf{B} / \partial t.\) In this expression \(\nabla \times \mathbf{E}\) is computed with \(t\) held fixed and \(\partial\mathbf{B} / \partial t\) is calculated with \((x,\,y,\,z)\) fixed. Use Stoke's Theorem to derive Faraday's law \[\oint_C \mathbf{E} \cdot d\mathbf{r} = - \frac{\partial}{\partial t} \iint_S \mathbf{B} \cdot \mathbf{n} \,d\sigma ,\] where \(C\) represents a wire loop through which current flows counterclockwise with respect to the surface's unit normal \(\mathbf{n},\) giving rise to the voltage \[\oint_C \mathbf{E}\cdot d\mathbf{r}\] around \(C.\) The surface integral on the right side of the equation is called the magnetic flux, and \(S\) is any oriented surface with boundary \(C.\)
6. Let \[\mathbf{F} = - \frac{GmM}{\lvert\mathbf{r}\rvert^3}\mathbf{r}\] be the gravitional force field defined for \(\mathbf{r} \ne \mathbf{0}.\) Use Gauss's law to show that there is no continuously differentiable vector field \(\mathbf{H}\) satisfying \(\mathbf{F} = \nabla \times \mathbf{H}.\)
7. Let \(S\) be an oriented surface parametrized by \(\mathbf{r}(u,\,v).\) Define the notation \(d\vec{\sigma} = \mathbf{r}_u \,du \times \mathbf{r}_v\,dv\) so that \(d\vec{\sigma}\) is a vector normal to the surface. Also, the magnitude \(d\sigma = \lvert d\vec{\sigma}\rvert\) is the element of surface area. Derive the identity \[d\vec{\sigma} = (EG -F^2 )^{1/2} \,du\,dv\] where \[E = \lvert \mathbf{r}_u \rvert^2, \,\,\, F = \mathbf{r}_u \cdot \mathbf{r}_v \,\,\, \text{and}\,\,\, G = \lvert\mathbf{r}_v\rvert^2 .\]
8. Show that the volume \(V\) of a region \(D\) in space enclosed by the oriented surface \(S\) with outward normal \(\mathbf{n}\) satisfies the identity \[V = \frac{1}{3} \iint_S \mathbf{r} \cdot \mathbf{n} \,d\sigma,\] where \(\mathbf{r}\) is the position vector of the point \((x,\,y,\,z)\) in \(D.\)

Advanced Problems.

1. Research the definition of the line integral of complex functions, and evaluate the complex line integral of \[f(z) = \frac{1}{z^m} ,\,\, m\in \mathbb{Z}\] along the curve parametrized by \[\gamma (t) = e^{it} = \cos t + i \sin t ,\,\, 0\le t \le 2\pi.\]
2. Research the exterior derivative of first order differential forms, and find \(d\omega\) where \[\omega = M\,dx + N\,dy\] and \[\mathbf{F} = M\,\mathbf{i} + N\,\mathbf{j},\] a vector field with continuous partial derivatives.
3. Describe generalized Stokes' theorem, and deduce Green's theorem from generalized Stokes' theorem.
4. Find \(d\omega\) for \(\omega = M\,dx + N\,dy + P\,dz\) and \(\mathbf{F} = M\,\mathbf{i} + N\,\mathbf{j} + P\,\mathbf{k},\) and deduce three dimensional Stokes' theorem from generalized Stokes' theorem.
5. Find \(d\omega\) for \(\omega = M\,dy\,dz + N\,dz\,dx + P\,dx\,dy\) and \(\mathbf{F} = M\,\mathbf{i} + N\,\mathbf{j} + P\,\mathbf{k},\) and deduce divergence theorem from generalized Stokes' theorem.
6. Deduce change of variable formula for multiple integrals from Stokes' theorem.
7. Research "Poincare's lemma"; describe Poincare's lemma in 3 dimensional spaces.
8. Research "partition of unity", and extend the definition of multiple integrals for the functions defined on arbitrary open sets.