We will cover following topics

Limits

Let $S$ be an open set containing $(x_{0}, y_{0}$ , and let $f$ be a function of two variables defined on $S$ , except possibly at $(x_{0}, y_{0})$ . The limit of $f (x, y)$ as $(x, y)$ approaches $(x_{0}, y_{0})$ is $L$ , denoted as:

lim_{(x, y) \to (x_{0}, y_{0})} f (x, y) = L

It means that given any $ε > 0$ , there exists $δ > 0$ such that for all $(x, y) \neq (x_{0}, y_{0})$ , if $(x, y)$ is in the open disk centered at $(x_{0}, y_{0})$ with radius $δ$ , then $| f (x, y) - L | < ε$ .

Continuity

Let a function $f (x, y)$ be defined on an open disk $B$ containing $(x_{0}, y_{0})$ . Then,

$f$ is continuous at $(x_{0}, y_{0})$ if $lim_{(x, y) \to (x_{0}, y_{0})} f (x, y) = f (x_{0}, y_{0})$
$f$ is continuous on $B$ if $f$ is continuous at all points in $B$ . If $f$ is continuous at all points in $R^{2}$ , we say that $f$ is continuous everywhere.

Partial Derivaties

Let $z = f (x, y)$ be a continuous function on an open set $S$ in $R^{2}$ , then

The partial derivative of $f$ with respect to $x$ is:

f_{x} (x, y) = lim_{h \to 0} \frac{f (x + h, y) - f (x, y)}{h}

The partial derivative of $f$ with respect to $y$ is:

f_{y} (x, y) = lim_{h \to 0} \frac{f (x, y + h) - f (x, y)}{h}

The second partial derivative of $f$ with respect to $x$ then x is:

\frac{\partial}{\partial x} (\frac{\partial f}{\partial x}) = \frac{\partial^{2} f}{\partial x^{2}} = {(f_{x})}_{x} = f_{x x}

The second partial derivative of $f$ with respect to $x$ then $y$ is:

\frac{\partial}{\partial y} (\frac{\partial f}{\partial x}) = \frac{\partial^{2} f}{\partial y \partial x} = (f_{x})_{y} = f_{x y}

The similar definitions can be extended to functions of more than two variables.

Theorem: Let $f$ be defined such that $f_{x y}$ and $f_{y x}$ are continuous on an open set $S$ . Then, for each point $(x, y)$ in $S$ , $f_{x y} (x, y) = f_{y x} (x, y)$ .

Maxima and Minima

Second Derivative Test: Let $z = f (x, y)$ be differentiable on an open set containing $P = (x_{0}, y_{0})$ and let $D = f_{x x} (x_{0}, y_{0}) f_{y y} (x_{0}, y_{0}) - f_{x y}^{2} (x_{0}, y_{0})$ .

If $D > 0$ and $f_{x x} (x_{0}, y_{0}) > 0$ , then $P$ is a relative minimum of $f$
If $D > 0$ and $f_{x x} (x_{0}, y_{0}) < 0$ , then $P$ is a relative maximum of $f$
If $D < 0$ , then $P$ is a saddle point of $f$
lf $D = 0$ , the test is inconclusive.

Lagrange’s Method Of Multipliers

Lagrange’s method of multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints. We can be provided with a single constraint or multiple constraints.

Case I: Single Constraint

Suppose we have an objective function of two variables with a single constraint.

maximize $f (x, y)$
subject to $g (x, y) = 0$

We first find the Lagrangian, which is given by

L (x, y, λ) = f (x, y) - λ g (x, y)

Then, we solve the below equation to find the stationary points:

\nabla_{x, y, λ} L (x, y, λ) = 0

We then calculate the value of $f (x, y)$ at stationary points to find its maximum value.

Case II: Multiple Constraints

Suppose we have an objective function of $n$ variables and $m$ constraints.

maximize $f (x_{1}, x_{2}, \dots x_{n})$
subject to
$g_{1} (x, y) = 0$ ,
$g_{2} (x, y) = 0$ , $\dots$
$g_{m} (x, y) = 0$

We first find the Lagrangian given by:

L (x_{1}, \dots, x_{n}, λ_{1}, \dots λ_{m}) = f (x_{1}, \dots, x_{n}) - \sum_{k = 1}^{m} λ_{k} g_{k} (x_{1}, \dots, x_{n})

and solve

\nabla_{x_{1}, \dots, x_{n}, λ_{1}, \dots, λ_{m}} L (x_{1}, \dots, x_{n}, λ_{1}, \dots λ_{m}) = 0

which implies that

\nabla f (x) - \sum_{k = 1}^{m} λ_{k} \nabla g_{k} (x) = 0

and

g_{1} (x) = g_{2} (x) = \dots = g_{m} (x) = 0

Example: Maximize $f (x, y) = x + y$ subject to the constraint $x^{2} + y^{2} = 1$ .

Solution: Here $f (x, y) = x + y$ , $g (x, y) = x^{2} + y^{2} - 1$ and therefore,

\begin{aligned} (1) & L (x, y, λ) & = f (x, y) + λ \cdot g (x, y) \\ (2) & = x + y + λ (x^{2} + y^{2} - 1) \end{aligned}

Now, we calculate the gradient:

\begin{aligned} (3) & \nabla_{x, y, λ} L (x, y, λ) & = (\frac{\partial L}{\partial x}, \frac{\partial L}{\partial y}, \frac{\partial L}{\partial λ}) \\ (4) & = (1 + 2 λ x, 1 + 2 λ, x^{2} + y^{2} - 1) \end{aligned}

Therefore:

\nabla_{x, y, λ} L (x, y, λ) = 0 \Leftrightarrow {\begin{cases} 1 + 2 λ x = 0 \\ 1 + 2 λ y = 0 \\ x^{2} + y^{2} - 1 = 0 \end{cases}

which implies that the stationary points of $L$ are:

(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, - \frac{1}{\sqrt{2}}), (- \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}, \frac{1}{\sqrt{2}})

Evaluating the objective function $f$ at these points yields

f (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) = \sqrt{2},

f (- \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}) = - \sqrt{2}

Thus, the constrained maximum is $\sqrt{2}$ and the constraied minimum is $- \sqrt{2}$ .

Jacobian

The Jacobian martix is the matrix of all first-order partial derivatives of a vector-valued function. If the matrix is a square matrix, both the matrix and its determinant are referred to as the Jacobian in literature.

Suppose $f (x) : R^{n} \to R^{m}$ , then the Jacobian matrix $J$ of $f$ is an $m \times n$ matrix as deined below:

J = [\begin{array}{ccc} \frac{\partial f}{\partial x_{1}} & \dots & \frac{\partial f}{\partial x_{n}} \end{array}] = [\begin{array}{ccc} \frac{\partial f_{1}}{\partial x_{1}} & \dots & \frac{\partial f_{1}}{\partial x_{n}} \\ ⋮ & ⋱ & ⋮ \\ \frac{\partial f_{m}}{\partial x_{1}} & \dots & \frac{\partial f_{m}}{\partial x_{n}} \end{array}],

or component wise, $J_{i j} = \frac{\partial f_{i}}{\partial x_{j}}$

Example 1: Consider the function $f : R^{2} \to R^{2}$ , with $(x, y) \mapsto (f_{1} (x, y), f_{2} (x, y))$ , given by

f ([\begin{matrix} x \\ y \end{matrix}]) = [\begin{matrix} f_{1} (x, y) \\ f_{2} (x, y) \end{matrix}] = [\begin{matrix} x^{2} y \\ 5 x + \sin y \end{matrix}]

Then, we have

f_{1} (x, y) = x^{2} y

and

f_{2} (x, y) = 5 x + \sin y

J_{f} (x, y) = [\begin{matrix} \frac{\partial f_{1}}{\partial x} & \frac{\partial f_{1}}{\partial y} \\ \frac{\partial f_{2}}{\partial x} & \frac{\partial f_{2}}{\partial y} \end{matrix}] = [\begin{matrix} 2 x y & x^{2} \\ 5 & \cos y \end{matrix}]

and the Jacobian determinant is equal to

\det (J_{f} (x, y)) = 2 x y \cos y - 5 x^{2}

Example 2: Polar-Cartesian Transformation

The transformation from polar coordinates $(r, φ)$ to Cartesian coordinates $(x, y)$ , is given by the function $F : R^{+} \times [0, 2 π) \to R^{2}$ with components:

x = r \cos φ

y = r \sin φ

Therefore, the Jacobian matrix of $f$ is given by:

J_{f} (r, φ) = [\begin{matrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial φ} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial φ} \end{matrix}] = [\begin{matrix} \cos φ & - r \sin φ \\ \sin φ & r \cos φ \end{matrix}]

and the Jacobian determinant is equal to $r$ . It is used to transform integrals between the two coordinate systems:

\iint_{F (A)} f (x, y) d x d y = \iint_{A} f (r \cos φ, r sir φ) r d r d φ

Example 3: Spherical-Cartesian Transformation

The transformation from spherical coordinates $(r, θ, φ)$ to Cartesian coordinates $(x, y, z)$ is given by the function $F : R^{+} \times [0, π] \times [0, 2 π) \to R^{3}$ with components:

$x = r \sin θ \cos φ$
$y = r \sin θ \sin φ$
$z = r \cos θ$

The Jacobian matrix for this coordinate change is given by:

J_{f} (r, θ, φ) = [\begin{matrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial θ} & \frac{\partial x}{\partial φ} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial θ} & \frac{\partial y}{\partial φ} \\ \frac{\partial z}{\partial r} & \frac{\partial z}{\partial θ} & \frac{\partial z}{\partial φ} \end{matrix}]

= [\begin{matrix} \sin θ \cos φ & r \cos θ \cos ϕ & - r \sin θ \sin φ \\ \sin θ \sin φ & r \cos θ \sin φ & r \sin θ \cos φ \\ \cos θ & - r \sin θ & 0 \end{matrix}]

and the Jacobian determinant is equal to $r^{2} \sin θ$ .

As an example, since $d V = d x d y d z$ , this determinant implies that the differential volume element $d V = r^{2} \sin θ d r d θ d φ$ .

PYQs

Limits

1) Find $lim_{(x, y) \to (0, 0)} \frac{x^{2} y}{x^{3} + y^{3}}$ if it exists.

[2011, 10M]

Continuity

1) Let $f : D (\subseteq R^{2}) \to R$ be a function and $(a, b) \in D$ . If $f (x, y)$ is continuous at $(a, b)$ , then show that the function $f (x, b)$ and $f (a, y)$ are continuous at $x = a$ and at $y = b$ respectively.

[2019, 10M]

2) Let $f (x, y) = {\begin{cases} \frac{2 x^{4} y - 5 x^{3} y^{2} + y^{5}}{{(x^{2} + y^{2})}^{2}}, & (x, y) \neq (0, 0) \\ 0 & , (x, y) = (0, 0) \end{cases}$
Find a $δ > 0$ such that $| f (x, y) - f (0, 0) | < 0.01$ whenever $\sqrt{x^{2} + y^{2}} < δ$ .

[2016, 15M]

Remarks: Original question has printing mistake

3) Examine the continuity for the function $f (x, y) = {\begin{array}{cc} \frac{x^{2} - x \sqrt{y}}{x^{2} + y} & , (x, y) \neq (0, 0) \\ 0 & , (x, y) = (0, 0) \end{array}$

[2015, 12M]

4) Define a function $f$ of two real variables in the plane by: $f (x, y) = {\begin{matrix} \frac{x^{3} \cos \frac{1}{y} + y^{3} \cos \frac{1}{x}}{x^{2} + y^{2}}, for x, y \neq 0 \\ 0, otherwise \end{matrix}$

Check the continuity and differentiability of $f$ at (0,0).

[2012, 12M]

5) Show that the function given below is not continuous at the origin.

f (x, y) = {\begin{cases} 0 if x y = 0 \\ 1 if x y \neq 0 \end{cases}

[2005, 12M]

Partial Derivatives

1) If

u = s i n^{- 1} \sqrt{\frac{x^{1 / 3} + y^{1 / 3}}{x^{1 / 2} + y^{1 / 2}}},

then show that $\sin^{2} u$ is homogeneous function of $x$ and $y$ of degree $- \frac{1}{6}$ . Hence show that

$x^{2} \frac{\partial^{2} u}{\partial x^{2}} + 2 x y \frac{\partial^{2} u}{\partial x \partial y} + y^{2} \frac{\partial^{2} u}{\partial y^{2}}$ = $\frac{\tan u}{12} (\frac{13}{12} + \frac{\tan^{2} u}{12})$

[2018, 12M]

2) Let

$f (x, y) = x y^{2}$ , if $y > 0$ and $f (x, y) = - x y^{2}$ , if $y \leq 0$

Determine which of $\frac{\partial f}{\partial x} (0, 1)$ and $\frac{\partial f}{\partial y} (0, 1)$ exists and which does not exist.

[2018, 12M]

3) If $f (x, y) = {\begin{cases} \frac{x y (x^{2} - y^{2})}{x^{2} + y^{2}}, (x, y) \neq (0, 0) \\ 0, (x, y) = (0, 0) \end{cases}$
Calculate $\frac{\partial^{2} f}{\partial x \partial y}$ and $\frac{\partial^{2} f}{\partial y \partial x}$ at (0,0).

[2017, 15M]

4) Compute $f_{x y} (0, 0)$ and $f_{y x} (0, 0)$ for the function $f (x, y) = {\begin{cases} \frac{x y^{3}}{x + y^{2}}, (x, y) \neq (0, 0) \\ 0, (x, y) = (0, 0) \end{cases}$
Also discuss the continuity of $f_{x y}$ and $f_{y x}$ at (0,0).

[2013, 15M]

5) If $f (x, y)$ is a homogeneous function of degree $n$ in $x$ and $y$ , and has continuous first and second order partial derivatives, then show that:-
i) $x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f$
ii) $x^{2} \frac{\partial^{2} f}{\partial x^{2}} + 2 x y \frac{\partial^{2} f}{\partial x \partial y} + y^{2} \frac{\partial^{2} f}{\partial y^{2}} = n (n - 1) f$ .

[2010, 10M]

6) Let $f : R^{2} \to R$ be defined as:

f (x, y) = {\begin{cases} \frac{x y}{\sqrt{x^{2} + y^{2}}}, (x, y) \neq (0, 0) \\ 0, (x, y) = (0, 0) \end{cases}

Is $f$ continuous at (0,0)? Compute partial derivatives of $f$ at any point $(x, y)$ , if it exists.

[2009, 20M]

7) If $x = 3 \pm 0.01$ and $y = 4 \pm 0.01$ with approximately what accuracy can you calculate the polar coordinates $r$ and $θ$ of the point $P (x, y)$ . Express your estimates as percentage changes of the value that $r$ and $θ$ have at the point (3,4).

[2009, 20M]

8) Prove that if $z = ϕ (y + a x) + ψ (y - a x)$ , then $a^{2} \frac{\partial^{2} z}{\partial y^{2}} - \frac{\partial^{2} z}{\partial x^{2}} = 0$ for any twice differentiable $ϕ$ and $ψ$ . Here, $a$ is a constant.

[2007, 15M]

9) If $z = x f (\frac{y}{x}) + g (\frac{y}{x})$ , show that $x^{2} \frac{\partial^{2} z}{\partial x^{2}} + 2 x y \frac{\partial^{2} z}{\partial x \partial y} + y^{2} \frac{\partial^{2} z}{\partial y^{2}} = 0$

[2006, 15M]

10) Let $R^{2} \to R$ be defined as $f (x, y) = \frac{x y}{\sqrt{(x^{2} + y^{2})}}, (x, y) \neq (0, 0) = 0$ , prove that $f_{x}$ and $f_{y}$ exist $(0, 0)$ but $f$ is not differentiable at $(0, 0)$ .

[2005, 12M]

11) If the function $f$ is defined by: $f (x, y) = {\begin{cases} \frac{x y}{x^{2} + y^{2}}, (x, y) \neq (0, 0) \\ 0, (x, y) = (0, 0) \end{cases}$ then show that $f$ posseses both partial derivative at origin but it is not continuous thereat.

[2004, 15M]

Maxima and Minima

1) Find the maximum and the minimum value of $x^{4} - 5 x^{2} + 4$ on the interval $[2, 3]$ .

[2018, 13M]

2) A conical tent is of given capacity. For the least amount of Canvas required for it, find the ratio of its height to the radius of its base.

[2015, 13M]

3) Which point of the sphere $x^{2} + y^{2} + z^{2} = 1$ is at the maximum distance from the point $(2, 1, 3)$ .

[2015, 13M]

4) Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius $a$ .

[2014, 15M]

5) Find the maximum or minimum values of $x^{2} + y^{2} + z^{2}$ subject to the condition $a x^{2} + b y^{2} + c z^{2} = 1$ and $l x + m y + n z = 0$ interpret result geometrically.

[2014, 20M]

6) Find the point of local extrema and saddle points of the function $f$ for two variable defined by $f (x, y) = x^{3} + y^{3} - 63 (x + y) + 12 x y$ .

[2012, 20M]

7) Show that a box (rectangular parallelepiped) of maximum volume $V$ with prescribed surface area is a cube.

[2010, 20M]

8) A space probe in the shape of the ellipsoid $4 x^{2} + y^{2} + 4 z^{2} = 16$ enters the earth’s atmosphere and its surface begins to heat. After one hour, the temperature at the point $(x, y, z)$ on the probe surface is given by $T (x, y, z) = 8 x^{2} + 4 y z - 16 z + 1600$ . Find the hottest point on the probe surface.

[2009, 20M]

9) Determine the maximum and minimum distances of the origin from the curve given by the equation $3 x^{2} + 4 x y + 6 y^{2} = 140$

[2008, 20M]

10) Find the maximum and minimum radii vectors of the section of the surface $(x^{2} + y^{2} + z^{2}) = a^{2} x^{2} + b^{2} y^{2} + c^{2} z^{2}$ by the plane $l x + m y + n z = 0$ .

[2001, 15M]

Lagrange’s Method of Multipliers

1) Find the maximum and minimum values of $x^{2} + y^{2} + z^{2}$ subject to the conditions $\frac{x^{2}}{4} + \frac{y^{2}}{5} + \frac{z^{2}}{25} = 1$ and $x + y - z = 0$ .

[2016, 20M]

2) Using Lagrange’s multiplier method find the shortest distance between the line $y = 10 - 2 x$ and the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{9} = 1$

[2013, 20M]

3) Find a rectangular parallelopiped of greatest volume for a give total surface area $S$ using Lagrange’s method of multipliers.

[2007, 20M]

4) A rectangular box open at the top is to have a volume of 4 $m^{3}$ . Using Lagrange’s method of multipliers, find the dimensions of the box so that the material of a given type required to construct it may be least.

[2003, 15M]

Jacobian

1) Using the Jacobian method, show that if $f^{'} (x) = \frac{1}{1 + x^{2}}$ and $f (0) = 0$ , then

f (x) + f (y) = f (\frac{x + y}{1 - x y})

[2019, 8M]

2) By using the transformation $x + y = u$ , $y = u v$ , evaluate the integral $\iint {x y (1 - x - y)}^{1 / 2} d x d y$ taken over the area enclosed by the straight lines $x = 0$ , $y = 0$ and $x + y = 1$ .

[2014, 15M]

3) If $u = x + y + z$ , $u v = y + z$ and $u v w = z$ , then find $\frac{\partial (x, y, z)}{\partial (u, v, w)}$

[2005, 15M]

4) Let the roots of the below equation in $λ$ be $u, v, w$ . $(λ - x)^{3} + (λ - y)^{3} + (λ - z)^{3} = 0$

Prove that: $\frac{\partial (u, v, w)}{\partial (x, y, z)} = - 2 \frac{(y - z) (z - x) (x - y)}{(u - v) (v - w) (w - u)}$

[2004, 15M]

5) If the roots of the equation $(λ - u)^{3} + (λ - v)^{3} + (λ - w)^{3} = 0$ in $λ$ are $x, y, z$ , show that:

\frac{\partial (x, y, z)}{\partial (u, v, w)} = - \frac{2 (u - v) (v - w) (w - u)}{(x - y) (y - z) (z - x)}

[2002, 15M]

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