Applications of single and several variable calculus (MTH112M 26-27)

Term
Semester I
Duration
Sep 22, 2026 - present
Instructor
Debasis Sen and Abhijit Pal
TA
Indrajit Ghosh

Room: L18

Class Timing: Tue 10-10:50

Course Description: Website

Office Hours: Wed 10:00 - 10:50 (Google Meet )

Textbooks and References

  1. George B. Thomas Jr., Maurice D. Weir & Joel R. Hass., Thomas Calculus, (15th edition).
  1. Robert G. Bartle & Donald R. Sherbert., Introduction to Real Analysis.

Lectures

Disclaimer: There are two versions of notes provided for each lecture. The Live Note is what I write on the board during class, while the Typeset Note (LaTeX) is the version I prepare beforehand to have on hand during the lecture, so it may be a bit more elaborate. Both are for reference only.

Lecture 1
Date: Sep 22, 2026
Lecture 2
Date: Sep 29, 2026
Lecture 3
Date: Oct 06, 2026
Lecture 4
Date: Oct 13, 2026
Lecture 5
Date: Oct 27, 2026
Lecture 6
Delivered by Sumit Kumar
Date: Nov 03, 2026
Lecture 7
Delivered by Suvankar Majee
Date: Nov 10, 2026
Lecture 8
Date: Nov 17, 2026

Assignments

Assignment 1: Polar Coordinates, Applications of Integrations (Click to view)
Original PDF
  1. Find the area of the region in the first quadrant bounded on the left by the \(y\)-axis, below by the curve \(x = 2\sqrt{y}\), above left by the curve \(x = (y - 1)^2\), and above right by the line \(x = 3 - y\).
  2. Sketch the graph of the following curves, given in polar coordinates, in the Cartesian plane:
    \[ r = \frac{\sin \theta}{2}, \quad r^2 = -\cos \theta, \quad r = e^{\theta/10}, \quad r = \frac{1}{2} + \sin \theta, \quad r = -|\cos \theta|. \]
    3. Angle θ
    Polar Curve
    Live θ Value:
    0.00 π
    Interval: [0, 2π]
  3. Find the area of the region that is inside the circle \(r = 3\sin\theta\) and also inside \(r = 1 + \sin\theta\).
  4. Find the intersection points between the curves \(r = \cos 2\theta\) and \(r = \sin 2\theta\).
  5. Find the lengths of the polar curves:
    1. \(r(\theta) = \frac{6}{1 + \cos\theta}, \quad 0 \leq \theta \leq \frac{\pi}{2}\),
    2. \(r(\theta) = \frac{a\sin^2\theta}{2}, \quad 0 \leq \theta \leq \pi, \quad a > 0\).
  6. (Non-rectifiable curve) Let \(g : [0, 1] \to [0, 1]\) be defined by
    \[ g(t) = \begin{cases} t \cos \left(\frac{\pi}{2t}\right), & t \neq 0, \\ 0, & t = 0. \end{cases} \]
    Show that \(g\) is continuous. Consider the curve \(\alpha : [0, 1] \longrightarrow \mathbb{R}^2, \alpha(t) = (t, g(t))\). Show that the curve is non-rectifiable i.e., it has infinite length.
    Hint: Recall that the length of a continuous curve \(\alpha : [0, 1] \to \mathbb{R}^2\) is defined by
    \[ L(\alpha) = \sup_{0 = s_0 < s_1 < \cdots < s_m = 1} \sum_{j=0}^{m-1} ||\alpha(s_j) - \alpha(s_{j+1})||, \quad (1) \]
    where the supremum is taken over all possible partitions \(0 = s_0 < s_1 < \cdots < s_m = 1\) of the interval \([0, 1]\) (no bound on \(m\)). For our specific example, take the partitions
    \[ P_n = \left\{0, \frac{1}{2n}, \frac{1}{2n - 1}, \ldots, \frac{1}{3}, \frac{1}{2}, 1\right\}, \quad n \in \mathbb{N}. \]
    Let \(\ell_n\) denote the length of the polygonal path inscribed in the curve using this partition. Prove that
    \[ \ell_n > \sum_{k=1}^{2n} \frac{1}{k} \]
    and deduce that the curve \(\alpha\) is not rectifiable.
  7. Consider the curve \(C\) defined by \(x(t) = \cos^3 t, y(t) = \sin^3 t, \quad 0 \leq t \leq \frac{\pi}{2}\).
    1. Find the length of the curve.
    2. Find the area of the surface generated by revolving \(C\) about the \(x\)-axis.
      3. 3D Surface of Revolution: Astroid about x-axis
      Sweep Angle φ:
      0.00 π
      Interval: [0, 2π]
  8. Let \(C\) denote the circular disc of radius \(b\) centered at \((a, 0)\) where \(0 < b < a\). Find the volume of the torus that is generated by revolving \(C\) around the \(y\)-axis using
    1. the Washer Method
    2. the Shell Method.
    9. 3D Torus Volume: Washer vs Shell Method
    Sweep Progress:
    y = -0.80
    Interval: y ∈ [-0.8, 0.8]
  9. A curved wedge is cut from a right circular cylinder of radius 3 by two planes. One plane is perpendicular to the axis of the cylinder and passes through the centre; the second plane also passes through the centre and meets the first plane at a \(45^\circ\) angle. Find the volume of the wedge (the region of the cylinder between the two planes).
    10. 3D Slicing Method: Volume of a Cylindrical Wedge

    This visualization matches your lecture notes. It shows the static, transparent wedge while the slicing right triangle sweeps across the \(x\)-axis to accumulate the volume.

    Cross-Section Slice:
    x = -3.00
    Interval: x ∈ [-3, 3]
  10. Using the theorem of Pappus, find the centroid of a thin, flat plate covering the region enclosed by the parabolas \(y = 2x^2\) and \(y = 3 - x^2\).
Assignment 2: Vectors, Curves, Surfaces, Vector Equations (Click to view)
Original PDF
  1. Consider the planes
    \[ x + y + z = 1, \] \[ 2x + ay + 3z = 4, \] \[ x + 2y + 2z = b, \]
    where \(a\) and \(b\) are parameters. Determine the values of \(a\) and \(b\) such that the three planes
    1. intersect at a single point,
    2. intersect in a line,
    3. intersect (taken two at a time) in three distinct parallel lines.
  2. Evaluate the distance between the lines
    \[ \frac{x - 2}{4} = \frac{y - 7}{-4} = \frac{z + 2}{3} \quad \text{and} \quad \frac{x + 1}{1} = \frac{y + 2}{4} = \frac{z - 1}{-3}. \]
  3. Show that the distance between the point \(Q = (x_0, y_0, z_0)\) and the plane
    \[ (x, y, z) \cdot \mathbf{n} = d \]
    is given by
    \[ \frac{|\mathbf{n} \cdot (x_0, y_0, z_0) - d|}{||\mathbf{n}||}. \]
    1. Find an equation for the surface consisting of all points that are equidistant from the point \((-1, 0, 0)\) and the plane \(x = 1\).
    2. Find an equation for the cylinder generated by a line through the curve
      \[ x^2 + y^2 = 4x, \quad z = 0 \]
      moving parallel to the vector \(\mathbf{i} + \mathbf{j} + \mathbf{k}\).
  5. Find the arc-length re-parametrization and then compute the curvature of the curve \(\alpha(t) = (e^t \cos t, e^t \sin t)\), \(t \in \mathbb{R}\).
  6. Let \(\alpha : (a, b) \to \mathbb{R}^2\) be a plane differentiable curve such that \(\alpha'(t) \neq 0\) for all \(t \in (a, b)\) and all its normals pass through a fixed point in \(\mathbb{R}^2\). Show that the image of \(\alpha\) lies on a circle.
  7. Show that the parabola \(y = ax^2\), \(a \neq 0\) has its largest curvature at its vertex and has no minimum curvature.
  8. Find the point on the curve \(c(t) = (5 \sin t)\vec{i} + (5 \cos t)\vec{j} + (12t)\vec{k}\) at a distance \(26\pi\) units from \((0, 5, 0)\) along the curve in the direction of increasing arc length.
Assignment 3: Functions of several variables (Click to view)
Original PDF
  1. Identify the points, if any, where the following functions fail to be continuous:
    1. \[ f(x, y) = \begin{cases} xy & \text{if } xy \ge 0 \\ -xy & \text{if } xy < 0 \end{cases} \] (Ans: Continuous everywhere)
    2. \[ f(x, y) = \begin{cases} xy & \text{if } xy \text{ is rational} \\ -xy & \text{if } xy \text{ is irrational.} \end{cases} \] (Ans: Discontinuous at points other than \(x\) axis and \(y\)-axis)
  2. Consider the function \(f : \mathbb{R}^2 \longrightarrow \mathbb{R}\) defined by \[ f(x, y) = \begin{cases} \frac{x^2y^2}{x^2y^2+(x-y)^2} & \text{if } (x, y) \neq (0, 0) \\ 0 & \text{if } (x, y) = (0, 0) \end{cases} \] Show that the function satisfy the following:
    1. The iterated limits \(\displaystyle \lim_{x \to 0} \left[ \lim_{y \to 0} f(x, y) \right]\) and \(\displaystyle \lim_{y \to 0} \left[ \lim_{x \to 0} f(x, y) \right]\) exist and equals 0;
    2. The function is continuous restricted to any line passing through origin,
    3. The limit \(\displaystyle \lim_{(x,y) \to (0,0)} f(x, y)\) does not exist. (So the function \(f(x, y)\) is not continuous at \((0, 0)\);)
    4. The partial derivatives of \(f\) exist at \((0, 0)\).
  3. Let \[ f(x, y) = \begin{cases} (x^2 + y^2) \sin \frac{1}{x^2+y^2} & \text{if } (x, y) \neq (0, 0) \\ 0 & \text{if } (x, y) = (0, 0). \end{cases} \] Show that \(f\) is differentiable at every point of \(\mathbb{R}^2\) but the partial derivatives are not continuous at \((0, 0)\).
  4. Let \(f(x, y) = |xy|\) for all \((x, y) \in \mathbb{R}^2\). Show that
    1. \(f\) is differentiable at \((0, 0)\).
    2. \(f_x(0, y_0)\) does not exist if \(y_0 \neq 0\).
  5. Suppose \(f : \mathbb{R}^2 \to \mathbb{R}\) is a function with \(f_x(x, y) = f_y(x, y) = 0\) for all \((x, y)\). Then show that \(f(x, y) = c\), a constant.
  6. Consider the function \(f : \mathbb{R}^2 \to \mathbb{R}\) defined by \[ f(x, y) = \begin{cases} \frac{x^2y\sqrt{|y|}}{x^4+y^2} & \text{if } (x, y) \neq (0, 0), \\ 0 & \text{if } (x, y) = (0, 0). \end{cases} \] Is \(f\) continuous at \((0, 0)\)? Do the directional derivatives of \(f\) at \((0, 0)\) exist along any direction? Is \(f\) differentiable at \((0, 0)\)? Justfy your answers.
  7. Consider the function \(F : \mathbb{R} \to \mathbb{R}^2,\quad F(t) = (\cos t, \sin t)\). Then show that there doest not exist \(0 < c < 2\pi\) such that \(F(2\pi) - F(0) = (2\pi - 0)F'(c)\).

    [Remark: Thus classical Mean Value Theorem (MVT) does not hold for vector-valued functions. But we always have \(||F(b) - F(a)|| \le (b - a) \sup_c ||F'(c)||\).]
Assignment 4 : Directional derivatives, Maxima, Minima, Lagrange Multipliers (Click to view)
Original PDF
  1. Let \(f : \mathbb{R}^2 \to \mathbb{R}\) be defined by \[ f(x, y) = \begin{cases} \frac{x^2y}{x^2+y^2} & \text{if } (x, y) \neq (0, 0) \\ 0 & \text{if } (x, y) = (0, 0). \end{cases} \] Show that the directional derivative of \(f\) at \((0, 0)\) in all directions exist but \(f\) is not differentiable at \((0, 0)\).
  2. Let \(f(x, y) = x^2e^y + \cos(xy)\). Find the directional derivative of \(f\) at \((1, 2)\) in the direction \((\frac{3}{5}, \frac{4}{5})\). (Ans: \(2e^2 - s \sin 2\))
  3. Find the equation of the surface generated by the normals to the surface \(x + 2yz + xyz^2 = 0\) at all points on the \(z\)-axis. (Ans: \(y = 2xz\))
  4. Examine the following functions for local maxima, local minima and saddle points:
    1. \(4xy - x^4 - y^4\) (Ans: \((0, 0)\) saddle, \((1, 1), (-1, -1)\) are local maxima)
    2. \(x^3 - 3xy^2\). (Ans: \((0, 0)\) saddle)
  5. Find the absolute maxima of \(f(x, y) = xy\) on the unit disc \(\{(x, y) : x^2 + y^2 \le 1\}\). (Ans: \(1/2\) at \((\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})\) and \((-\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}})\))
  6. Assume that among all rectangular boxes with fixed surface area of 20 square meters, there is a box of largest possible volume. Find its dimensions. (Ans: \(x = y = z = \sqrt{10/3}\))
  7. Consider the function \[ f(x, y) = x^2 - xy + y^2 - y. \] Find the direction (unit vector) \(\mathbf{u}\) and the values of directional derivative \(D_{\mathbf{u}}f(1, -1)\) which
    1. \(D_{\mathbf{u}}f(1, -1)\) is largest, (Ans: \((3/5, -4/5)\) and value is \(5\))
    2. \(D_{\mathbf{u}}f(1, -1)\) is smallest, (Ans: \((-3/5, 4/5)\) and value is \(-5\))
    3. \(D_{\mathbf{u}}f(1, -1) = 0\). (Ans: \(\pm(4/5, 3/5)\) and value is \(0\))
  8. A company produces steel boxes at three different plants in amounts \(x\), \(y\) and \(z\), respectively, producing an annual revenue of \(R(x, y, z) = 8xyz^2 - 200(x+y+z)\). The company is to produce 100 units annually. How should production be distributed to maximize revenue?
    (Ans: \((25, 25, 50)\))
  9. Find the point on the line of intersection of two planes \(3x + 4y + 5z = 0\) and \(2x + y + 3z = 5\) that is nearest to the origin.
    (Ans: \((5/3, -10/3, 5/3)\))
  10. Minimize the quantity \(x^2 + y^2 + z^2\) subject to the constraints \(x + 2y + 3z = 6\) and \(x + 3y + 9z = 9\).
    (Ans: \(369/59\))
  11. Let \(f(x, y) = \frac{1}{2}\left( \big||x|-|y|\big| - |x| - |y| \right)\). Is \(f\) continuous at \((0, 0)\)? Which directional derivatives of \(f\) exist at \((0, 0)\)? Is \(f\) differentiable at \((0, 0)\)?
    (Ans: Not continuous, directional derivatives does not exist in all direction, so not differentiable.)
Assignment 5 : Double integrals, change of variables (Click to view)
Original PDF
  1. Evaluate the following iterated integrals:
    1. \(\displaystyle \int_0^1 \int_0^{\sqrt{1-x^2}} \sqrt{1 - y^2}\, dy\, dx\), (Ans: \(2/3\))
    2. \(\displaystyle \int_0^\pi \int_x^\pi \frac{\sin y}{y}\, dy\, dx\), (Ans: \(2\))
    3. \(\displaystyle \int_0^1 \int_y^1 x^2 e^{xy}\, dx\, dy\). (Ans: \((e - 2)/2\))
  2. Evaluate \(\displaystyle \iint_R x\, dx\, dy\) where \(R\) is the region \(1 \le x(1 - y) \le 2\) and \(1 \le xy \le 2\). (Ans: \(1\))
  3. Using double integral, find the area enclosed by the curve \(r = \sin 3\theta\) given in polar coordinates. (Ans: \(\pi/4\))
  4. Compute \(\displaystyle \lim_{a \to \infty} \iint_{D(a)} e^{-(x^2+y^2)}\, dx\, dy\), where
    1. \(D(a) = \{(x, y) : x^2 + y^2 \le a^2\}\) (Ans: \(\pi\))
    2. \(D(a) = \{(x, y) : 0 \le x \le a,\ 0 \le y \le a\}\). (Ans: \(\pi/4\))
    Hence prove that
    \[ \text{(i) } \int_0^\infty e^{-x^2}\, dx = \frac{\sqrt{\pi}}{2}, \qquad \text{(ii) } \int_0^\infty x^2 e^{-x^2}\, dx = \frac{\sqrt{\pi}}{4}. \]
  5. Find the volume of the solid which is common to the cylinder \(x^2 + y^2 = 1\) and \(x^2 + z^2 = 1\). (Ans: \(16/3\))
  6. Using the change of variables \(u = x + y\) and \(v = x - y\), show that
    \[ \int_0^1 \int_0^x (x - y)\, dy\, dx = \frac{1}{2} \int_0^1 \int_v^{2-v} v\, du\, dv \]
    and then find the value of the integral. (Ans: \(\frac{1}{6}\))
Assignment 6 : Triple integral, Surface and Line integral (Click to view)
Original PDF
  1. Evaluate the integral
    \[ \iiint_W \frac{dz\, dy\, dx}{\sqrt{1 + x^2 + y^2 + z^2}}; \]
    where \(W\) is the ball \(x^2 + y^2 + z^2 \le 1\).
    (Ans: \(2\pi(\sqrt{2} - \ln(1 + \sqrt{2}))\))
  2. What is the integral of the function \(x^2z\) taken over the entire surface of a right circular cylinder of height \(h\) which stands on the circle \(x^2 + y^2 = a^2\)?
    What is the integral of the given function taken throughout the volume of the cylinder?
    (Ans: \(\frac{ha^4\pi}{4} + \frac{a^3h^2\pi}{2}\), \(\frac{h^2\pi a^4}{8}\).)
  3. Find the area of the surface \(x = uv\), \(y = u+v\), \(z = u-v\), where \((u, v) \in D = \{(u, v) \mid u^2+v^2 \le 1\}\).
    (Ans: \((\sqrt{18} - 4)/3\))
  4. Find the line integral of the vector field \(\mathbf{F}(x, y, z) = y\vec{i} - x\vec{j} + \vec{k}\) along the path \(\mathbf{c}(t) = (\cos t, \sin t, \frac{t}{2\pi})\), \(0 \le t \le 2\pi\) joining \((1, 0, 0)\) to \((1, 0, 1)\).
    (Ans: \(1 - 2\pi\))
  5. Evaluate \(\displaystyle \int_C \mathbf{T} \cdot d\mathbf{R}\), where \(C\) is the circle \(x^2 + y^2 = 1\) and \(\mathbf{T}\) is the unit tangent vector.
    (Ans: \(2\pi\))
  6. Show that the integral \(\displaystyle \int_C yz\, dx + (xz + 1)\, dy + xy\, dz\) is independent of the path \(C\) joining \((1, 0, 0)\) and \((2, 1, 4)\).
  7. Let \(C\) be the upper semicircular arc
    \[ x^2 + y^2 = 1, \quad z = 0, \quad y \ge 0, \]
    oriented from \((1, 0, 0)\) to \((-1, 0, 0)\). Show that:
    1. \[ \int_C 2xy\, dx + x^2\, dy + 3z^2\, dz = 0. \]
    2. \[ \int_C -y\, dx + x\, dy = \pi. \]
Assignment 7 : Green’s /Stokes’ /Gauss’ Theorems (Click to view)
Original PDF
  1. Use Green’s Theorem to compute \(\displaystyle \int_C (2x^2 - y^2)\, dx + (x^2 + y^2)\, dy\) where \(C\) is the boundary of the region \(\{(x, y) : x, y \ge 0,\ x^2 + y^2 \le 1\}\).
  2. Use Stokes’ Theorem to evaluate the line integral \(\displaystyle \int_C -y^3\, dx + x^3\, dy - z^3\, dz\), where \(C\) is the intersection of the cylinder \(x^2 + y^2 = 1\) and the plane \(x + y + z = 1\) and the orientation of \(C\) corresponds to counterclockwise motion in the \(xy\)-plane.
  3. Let \(S\) be the unit sphere \(x^2 + y^2 + z^2 = 1\). Evaluate the following surface integral using Divergence Theorem:
    \[ \iint_S \left[ x(2x + 3e^{z^2}) + y(-y - e^{x^2}) + z(2z + \cos^2 y) \right] d\sigma. \]
  4. Let \(\vec{F} = \frac{\vec{r}}{|\vec{r}|^3}\) where \(\vec{r} = x\vec{i} + y\vec{j} + z\vec{k}\) and let \(S\) be any surface that surrounds the origin. Prove that
    \[ \iint_S \vec{F} \cdot \mathbf{n}\, d\sigma = 4\pi. \]
  5. Let \(D\) be the solid bounded by \(z = 0\) and the paraboloid \(z = 4 - x^2 - y^2\). Let \(S\) be the boundary of \(D\). If \(\mathbf{F}(x, y, z) = (x^3 + \cos(yz),\, y^3,\, x + \sin(xy))\), use divergence theorem to evaluate
    \[ \iint_S \mathbf{F} \cdot \mathbf{n}\, d\sigma \]
    where \(\mathbf{n}\) is the outward normal to the surface \(S\).
  6. Verify Stokes’ theorem for the vector field \(\mathbf{F} = (-y, x, z)\) and the surface \(S\) given by the upper hemisphere \(x^2 + y^2 + z^2 = 1\), \(z \ge 0\), oriented upward. The boundary curve \(C\) is oriented counterclockwise when viewed from above.
  7. Let \(\mathbf{F} = (x^3, y^3, z^3)\) and let \(V\) be the solid unit ball \(x^2 + y^2 + z^2 \le 1\). Using Gauss’ divergence theorem, evaluate the outward flux
    \[ \iint_{\partial V} \mathbf{F} \cdot \mathbf{n}\, d\sigma. \]
  8. Use Green’s theorem to evaluate \(\displaystyle \oint_C (x^2 - y^2)\, dx + 2xy\, dy\), where \(C\) is the positively oriented boundary of the region \(R\) enclosed by the circle \(x^2 + y^2 = 1\).

Last Updated: Oct 28, 2026