Fall 2025: Math 147 Homework

MW: Section 14.3, #3, 5, 13, 15, 37, 43.

OS: Section 4.2, # 61, 62, 63, 66, 74, 80, 81.

1. Use polar coordinates to analyze the following limits:

1. (i) \(\lim_{(x,y)\to(0,0)} \dfrac{x^2-y^2}{x^2+y^2}\).
2. (ii) \(\lim_{(x,y)\to(0,0)} \dfrac{x^3+x^5}{x^2+y^2}\).

2. Determine the value of the constant \(c\) so that \( f(x,y) = \begin{cases} \dfrac{x^3+xy^2+2x^2+2y^2}{x^2+y^2}, & \text{if } (x,y)\ne(0,0) \\ c, & \text{if } (x,y)=(0,0) \end{cases} \) is a continuous function.

3. OS: Section 4.2, # 102, 110, 111.

4. For \(F(x,y,z) = (x^2+y^2+z^2,\ 3xyz,\ \cos(x)+\sin(y)+e^z)\), calculate \(\lim_{(x,y,z)\to(1,-1,1)} F(x,y,z)\).

MW, Section 15.1: # 13-41, every other odd problem, and # 53.

2. For \( f(x,y) = \begin{cases} \dfrac{3x^2y-y^3}{x^2+y^2}, & \text{if } (x,y)\ne(0,0) \\ 0, & \text{if } (x,y)=(0,0), \end{cases} \) find formulas for \(f_x(x,y)\) and \(f_y(x,y)\). Note: You can just take partial derivatives as usual when \((x,y)\ne(0,0)\), and then use the limit definition to find what the partial derivatives are when \((x,y)=(0,0)\).

OS, Section 4.4: Find the tangent lines in the \(x\) and \(y\) directions for the functions and points given in # 171, 173, 176. Then try to find the tangent planes, using the corresponding tangent vectors. Also: Use the limit definition to show that \(f(x,y) = 3x^2+y\) is differentiable at \((1,-1)\). Then try showing \(f(x,y)\) is differentiable at any point \((a,b)\).

OS, Section 4.4: # 179, 191 and: (i) Use the definition of differentiability to show that \(2x^2+3y\) is differentiable at all \((a,b)\in \mathbb{R}^2\) and (ii) Determine whether or not the function \( f(x,y) = \begin{cases} \dfrac{x^5}{x^2+y^2}, & \text{if } (x,y)\ne(0,0) \\ 0, & \text{if } (x,y)=(0,0) \end{cases} \) is differentiable at \((0,0)\).

1. For the function \( f(x,y) = \begin{cases} \dfrac{2x^2y^2}{\sqrt{x^2+y^2}}, & \text{if } (x,y)\ne(0,0) \\ 0, & \text{if } (x,y)=(0,0) \end{cases} \), use the definition to show that \(f(x,y)\) is differentiable at \((0,0)\). Then verify that both partial derivatives are continuous at \((0,0)\).

2. Use the definition to verify that \(f(x,y,z) = xyz+75\) is differentiable at all points \((a,b,c)\in \mathbb{R}^3\).

Find \(DF(2,3,1)\) for the function \(F(x,y,z) = (x^2y^3z,\ e^{xy^2z^3},\ \cos(xyz))\) and OS, Section 4.7: # 311-339, every other odd. Just find the critical points, don't classify them.

In addition, use an \(\epsilon\), \(\delta\) argument to show that, given a function \(f(x,y)\), a point \((a,b)\) in its domain, and \(L\in\mathbb{R}\), the statements \(\lim_{(x,y)\to(a,b)} f(x,y) = L\) and \(\lim_{(x,y)\to(a,b)} |f(x,y)-L| = 0\) are equivalent, i.e., each statement implies the other statement.

OS, Section 4.7: Classify the critical points you found in # 319-339, every other odd, in the previous assignment.

MW, Section 16.3: # 21, 24, 27, 32, 34, and OS, Section 4.7, # 346, 347, 348.

1. Find and classify the critical points for: \(f(x,y,z) = x^2-xy+z^2-2xz+6z\) and \(g(x,y,z) = xy+xz+2yz+\dfrac{1}{x}\).

2. OS, Section 4.5: # 215, 217, 219, 243, 244, 254.

MW, Section 16.1: # 21, 22, 27, 33; And: Use the limit definition to find the directional derivative of \(f(x,y) = 3x^2+2xy+5\) at \((1,2)\) in the direction of \(\cos(\tfrac{\pi}{3})\vec{i}+\sin(\tfrac{\pi}{3})\vec{j}\), then verify your answer using the gradient formula.

This homework problem is Bonus Problem 4, to be turned in on Friday, September 19 for a maximum of three bonus points. In class we noted that iterated limits need not be equal, for functions of two variables. The failure of the equality of the limits \(\lim_{k\to a}\lim_{h\to b} L(h,k)\) and \(\lim_{h\to b}\lim_{k\to a} L(h,k)\) for \(L(h,k) = \dfrac{h+k}{h-k}\) is related to the failure of \(\lim_{(h,k)\to(0,0)} L(h,k)\) to exist. Here is a sufficient condition:

1. For \(f(x,y) = \dfrac{x^2}{x^2+y^2}\), show that \(\lim_{(x,y)\to(0,0)} f(x,y)\) does not exist, while each of \(\lim_{y\to 0}\lim_{x\to 0} f(x,y)\) and \(\lim_{x\to 0}\lim_{y\to 0} f(x,y)\) exist, but are not equal.

2. For \(f(x,y) = \dfrac{x^2+y+1}{x+y^2+1}\), show that \(\lim_{(x,y)\to(0,0)} f(x,y)\), \(\lim_{y\to 0}\lim_{x\to 0} f(x,y)\), \(\lim_{x\to 0}\lim_{y\to 0} f(x,y)\) exist and are all equal.

3. For \( f(x,y) = \begin{cases} 1, & \text{if } xy\ne 0 \\ 0, & \text{if } xy=0 \end{cases} \) show that \(\lim_{y\to 0}\lim_{x\to 0} f(x,y) = 1 = \lim_{x\to 0}\lim_{y\to 0} f(x,y)\), but \(\lim_{(x,y)\to(0,0)} f(x,y)\) does not exist.

OS, Section 4.8: # 361-366.

Let \(S\) be the surface that is the graph of the equation \(z=f(x,y)\) and suppose that \(P=(a,b,f(a,b))\) is a point on \(S\). Let \(L_0\) be a line in \(\mathbb{R}^2\) passing through \((a,b)\) and \(C\) denote the curve consisting of the points on \(S\) lying above \(L_0\). Let \(\vec{u} = u_1\vec{i}+u_2\vec{j}\) be a unit direction vector for \(L_0\). Give a rigorous explanation for why

is the parametric equation of the line tangent to \(C\) at the point \(P\). We will assume that \(f(x,y)\ge 0\) in an open disk about \((a,b)\) (so the surface lies above the \(xy\)-plane near \(P\)) and the first order partials of \(f(x,y)\) exist and are continuous in an open disk about \((a,b)\). Due Friday, September 26. (4 points)

OS, Section 4.8: # 377, 379, 382, 384, 387.

OS, Section 5.1: # 13, 19, 21, 25, 37, 30.

MW, Section 17.2: # 7-19, odd.

Work the following problem for three bonus points and turn in your solution on Friday, October 3. Suppose \(a(t)\) is a function of one variable, and \(f(x,y) = a(x)a(y)\). Let \(R\) denote the square \([c,d]\times[c,d]\). Prove that \(\displaystyle\int\!\!\int_R f(x,y)\, dA = \left(\int_c^d a(x)\, dx\right)^2\).

OS, Section 5.3: # 149, 154, 155, 158, 159.

OS, Section 5.7: # 388, 389, 392, 398.

OS, Section 5.7: # 390, 394, 397.

Suppose \(T(u,v) = (au+bv, cu+dv)\) is a linear transformation from the \(uv\)-plane to the \(xy\)-plane. Give a good proof that \(T\) is one-to-one if and only if \(ad-bc\) is not zero. This problem is due in class on Wednesday October 15 and is worth 5 points. Hint: For one direction, you will end up solving a system of two homogeneous equations in two unknowns.

OS, Section 5.6: # 391, 396. Note that these problems give the inverse transformation.

Calculate the following improper integrals.

1. (i) \(\displaystyle\int\!\!\int_D \dfrac{1}{\sqrt{xy}}\, dA\), for \(D = [0,1]\times[0,1]\).
2. (ii) \(\displaystyle\int\!\!\int_D \ln\sqrt{x^2+y^2}\, dA\), for \(D = \{0\le x^2+y^2\le 1\}\).
3. (iii) \(\displaystyle\int\!\!\int_D \dfrac{1}{x^2y^3}\, dA\), for \(D = [1,\infty)\times[1,\infty)\).

MW, Section 17.4: # 1, 5, 9, 16.

OS, Section 5.4: # 193, 194, 211, 212, 231.

Section 5.5: # 253-256, 269, 270, 271.

1. Calculate \(\displaystyle\int\!\!\int\!\!\int_B 3x+y+z^2\, dV\) where \(B\) is the solid parallelepiped spanned by the vectors \(v_1 = (1,1,1)\), \(v_2 = (1,2,1)\), \(v_3 = (2,2,2)\).

2. Let \(B_0\) denote the solid cube in the \(uvw\)-coordinate system centered at the origin with sides of length 2. Let \(B\) denote the solid box in the \(xyz\)-coordinate system centered at \((1,-2,1)\) whose sides have lengths 2, 4, 6 in the \(x,y,z\) directions. Find a transformation \(G(u,v,w)\) from the \((u,v,w)\)-coordinate system to the \(xyz\)-coordinate system taking \(B_0\) to \(B\).

3. For \(B\) as in problem 2 above, use the transformation you found to calculate \(\displaystyle\int\!\!\int\!\!\int_B xyz\, dV\).

To be turned in Friday during class.

For a \(3\times 3\) matrix \(A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}\), define \(A_{ij}\), for \(i\ne j\), to be the \(2\times 2\) matrix obtained by deleting the \(i\)th row and \(j\)th column of \(A\). We can define the determinant of \(A\) by expanding along any row or any column, according to the following formulas. In the formulas below, we use \(|C|\) to denote the determinant of the matrix \(C\), so that, in the present situation, \(|-|\) does not mean absolute value.

Now let \(A\) denote the matrix \(\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}\).

1. Use the formulas above to show that \(|A|\) is the same when expanding along the third row or expanding along the second column. (2 points)

3. Show that \(|A| = |A^t|\), where \(A^t\) denotes the transpose of \(A\), i.e., \(A^t = \begin{pmatrix} a & d & g \\ b & e & h \\ c & f & i \end{pmatrix}\). (3 points)

OS, Section 3.2: # 41-55, odd.

Give a proof of the differentiability properties (4)-(6) from today's lecture. Each part is worth 2 points, and this is due Monday, November 3.

OS, Section 3.3: # 102, 106, 107, 110 and MW, Section 18.1: # 29, 30, 32, 34.

OS, Section 6.2: # 75, 86, 92, 94 and the following problem: Let \(C\) be the curve with parametrization \(\vrt = (\cos(t), \sin(t), t)\), \(0\le t\le 2\pi\) so that \(C\) is that portion of the helix of radius one from \((1,0,0)\) to \((1,0,1)\). Find a second parametrization of \(C\) and use this to create a re-parametrization of \(C\). Then check that \(\int_C x+y+z\, ds\) is independent of the two parameterizations.

1. For the sphere \(S: x^2+y^2+z^2=4\), find the plane tangent to \(S\) at \(P=(1,1,\sqrt{2})\).

2. Let \(S\) denote the surface that is the graph of the function \(z=f(x,y)\). In terms of \(x,y,z\), find the equation of the plane tangent to \(S\) at the point \(P=(x_0,y_0,z_0)\).

3. Find a parameterization in terms of \(u,v\) for the plane you found in problem 1.

Look up in any calculus book the definition of \(\vec{v}\times\vec{w}\) for vectors \(\vec{v},\vec{w}\in \mathbb{R}^3\). Read and then write down proofs of the following facts: (a) \(\vec{v}\times\vec{w}\) is orthogonal to the plane spanned by \(\vec{v}\) and \(\vec{w}\) (assuming these vectors are not collinear) and (b) the length of \(\vec{v}\times\vec{w}\) equals the area of the parallelogram spanned by \(\vec{v}\) and \(\vec{w}\). This problem is worth 5 points and is due in class on Monday, November 10.

1. Calculate \(\displaystyle\int\!\!\int_S \sqrt{x^2+y^2+1}\, dS\), where \(S\) is the helicoid given parametrically by \(G(r,\theta) = (r\cos\theta, r\sin\theta, \theta)\), with \(0\le r\le 1\) and \(0\le \theta\le 2\pi\). What is the surface area of \(S\)?

2. Let \(S\) denote the unit cube in the first octant of \(\mathbb{R}^3\) spanned by \(\vec{e}_1, \vec{e}_2, \vec{e}_3\). Calculate \(\displaystyle\int\!\!\int_S xyz\, dS\). Hint: There are six separate surface integrals to calculate, but three of them have an obvious answer (with a little thought).

OS Section 6.2: # 68, 69, 70 and OS Section 6.6: # 303, 304, 305, 309.

1. Suppose \({\bf F} = x\vi+y\vj+(z-2)\vk\). Calculate \(\displaystyle\int\!\!\int_S {\bf F}\cdot d{\bf S}\), for \(S\) the helicoid with parameterization \(G(u,v) = (u\cos(v), u\sin(v), v)\), with \(0\le u\le 1\) and \(0\le v\le 2\pi\).

2. Let \({\bf F} = x\vi+y\vj+z\vk\) and \(S\) denote the sphere of radius \(R\) centered at the origin. Calculate \(\displaystyle\int\!\!\int_S {\bf F}\cdot d{\bf S}\) with respect to the outward normal in two ways: First by parameterizing \(S\) and second, without parameterizing \(S\), i.e., by just thinking about the situation.

Verify the Divergence Theorem for \({\bf F} = x^2\vi+y^2\vj+z^2\vk\), and \(B\) the solid rectangle \(0\le x\le a\), \(0\le y\le b\), \(0\le z\le c\).

1. Calculate \(\displaystyle\int\!\!\int_S {\bf F}\cdot d{\bf S}\) with respect to the outward normal, for the vector field \({\bf F} = yz^3\vi+e^{x^2+z^2}\vj+\cos(\sqrt{x^2+y^2})\vk\) and \(S\) the torus obtained by revolving the circle \((y-3)^2+z^2=4\) in the \(yz\)-plane about the \(y\)-axis.

A parametrization for \(S\) is: \(x = (3+2\cos(v))\sin(u)\), \(y = (3+2\cos(v))\cos(u)\), \(z = 2\sin(v)\), with \(0\le u,v\le 2\pi\).

2. Verify the Divergence Theorem for \({\bf F} = (x-y)\vec{i}+(x+z)\vec{j}+(z-y)\vec{k}\), for the surface that consists of the cone \(x^2+y^2=z^2\), \(0\le z\le 1\) with a circular top at the \(z=1\) level.

1. Verify Green's Theorem for \({\bf F} = (x^2y+x)\vec{i}+(y^3-xy^2)\vec{j}\) and \(D\) the region bounded by the circles \(x^2+y^2=9\) and \(x^2+y^2=4\). Note that \(\partial D\) has an inner component and outer component. These must be oriented correctly so that the region \(D\) remains on the left as one travels along \(\partial D\).

2. Use a line integral to find the area contained in the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\).

Set \({\bf F} = z^2\vec{i}+x^2\vec{j}-y^2\vec{k}\).

1. (i) Calculate \(\nabla\times{\bf F}\).
2. (ii) Let \(C\) be the square path with sides equal to \(a\) centered at the point \((x_0,0,z_0)\) lying in the \(xz\)-plane oriented so that each side is parallel to the \(x\) or \(z\) axis. Calculate \(\int_C {\bf F}\cdot d{\bf r}\).
3. (iii) Divide your answer in (ii) by the area of the square and take the limit as \(a\) goes to zero.
4. (iv) Use your answer in (i) to corroborate your answer in (iii).

2. Verify Stokes' Theorem for \({\bf F} = (-y, 2x, x+z)\) and \(S\) the upper hemisphere of the sphere of radius \(R\) centered at the origin.

OS, section 6.7: # 335, 339, 343.