Functions of Two or Three Variables
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:
\[\qquad \lim _{(x, y) \rightarrow (x_0, y_0)} f(x, y)=L\]It means that given any \(\varepsilon>0\), there exists \(\delta>0\) 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 \(\delta\), then \(\vert f(x, y)-L \vert < \varepsilon\).
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 \(\lim_{(x, y) \rightarrow (x_0, y_0)} f(x, y)=f(x_0, y_0)\)
- \(f\) is continuous on \(B\) if \(f\) is continuous at all points in \(B\). If \(f\) is continuous at all points in \(\mathbb{R}^{2}\), 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:
- The partial derivative of \(f\) with respect to \(y\) is:
- The second partial derivative of \(f\) with respect to \(x\) then x is:
- The second partial derivative of \(f\) with respect to \(x\) then \(y\) is:
The similar definitions can be extended to functions of more than two variables.
Theorem: Let \(f\) be defined such that \(f_{xy}\) and \(f_{yx}\) 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
\[\mathcal{L}(x, y, \lambda) =f(x, y)-\lambda g(x, y)\]Then, we solve the below equation to find the stationary points:
\[\nabla_{x, y, \lambda} \mathcal{L}(x, y, \lambda)=0\]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 \(f(x_1, x_2, \cdots x_n)\)
subject to
\(g_1(x,y)=0\),
\(g_2(x,y)=0\), \(\cdots\)
\(g_m(x,y)=0\)
We first find the Lagrangian given by:
\[\mathcal{L}(x_1, \cdots, x_n, \lambda_1, \cdots \lambda_m) = f(x_1, \cdots, x_n) - \sum_{k=1}^m \lambda_k g_k(x_1, \cdots, x_n)\]and solve
\[\nabla_{x_1, \cdots, x_n, \lambda_1, \cdots, \lambda_m} \mathcal{L}(x_1, \cdots, x_n, \lambda_1, \cdots \lambda_m)=0\]which implies that
\[\nabla f(x) - \sum_{k=1}^m \lambda_k \nabla g_k(x)=0\]and
\[g_1(x)= g_2(x)= \cdots = g_m(x)=0\]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,
\[\begin{align} \mathcal{L}(x, y, \lambda) &=f(x, y)+\lambda \cdot g(x, y) \\ &=x+y+\lambda\left(x^{2}+y^{2}-1\right) \end{align}\]Now, we calculate the gradient:
\[\begin{align} \nabla_{x, y, \lambda} \mathcal{L}(x, y, \lambda) &=\left(\dfrac{\partial \mathcal{L}}{\partial x}, \dfrac{\partial \mathcal{L}}{\partial y}, \dfrac{\partial \mathcal{L}}{\partial \lambda}\right) \\ &=(1+2 \lambda x, 1+2 \lambda, x^2+y^2-1) \end{align}\]Therefore:
\[\nabla_{x, y, \lambda} \mathcal{L}(x, y, \lambda)=0 \Leftrightarrow \left\{\begin{array}{l}{1+2 \lambda x=0} \\ {1+2 \lambda y=0} \\ {x^{2}+y^{2}-1=0}\end{array}\right.\]which implies that the stationary points of \(\mathcal{L}\) are:
\[\left(\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2},-\dfrac{1}{\sqrt{2}}\right), \left(-\dfrac{\sqrt{2}}{2},-\dfrac{\sqrt{2}}{2}, \dfrac{1}{\sqrt{2}}\right)\]Evaluating the objective function \(f\) at these points yields
\[f\left(\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right)=\sqrt{2},\] \[f\left(-\dfrac{\sqrt{2}}{2},-\dfrac{\sqrt{2}}{2}\right)=-\sqrt{2}\]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 \(f(x) : \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\), then the Jacobian matrix \(J\) of \(f\) is an \(m \times n\) matrix as deined below:
\[\mathbf{J} = \left[\begin{array}{ccc}{\dfrac{\partial \mathbf{f}}{\partial x_{1}}} & {\cdots} & {\dfrac{\partial \mathbf{f}}{\partial x_{n}}}\end{array}\right] = \left[\begin{array}{ccc}{\dfrac{\partial f_{1}}{\partial x_{1}}} & {\cdots} & {\dfrac{\partial f_{1}}{\partial x_{n}}} \\ {\vdots} & {\ddots} & {\vdots} \\ {\dfrac{\partial f_{m}}{\partial x_{1}}} & {\cdots} & {\dfrac{\partial f_{m}}{\partial x_{n}}}\end{array}\right],\]or component wise, \(\mathbf{J}_{i j}= \dfrac{\partial f_{i}}{\partial x_{j}}\)
Example 1: Consider the function \(\mathbf{f} : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\), with \((x, y) \mapsto \left(f_{1}(x, y), f_{2}(x, y)\right)\), given by
\[f \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} f_1(x,y) \\ f_2(x,y) \end{bmatrix} = \begin{bmatrix} x^2y \\ 5x+ \sin y \end{bmatrix}\]Then, we have
\[f_1(x,y)=x^2y\]and
\[f_2(x,y)=5x + \sin y\] \[\mathbf{J}_{f}(x, y) = \begin{bmatrix} \dfrac{\partial f_{1}}{\partial x} & \dfrac{\partial f_{1}}{\partial y} \\ \dfrac{\partial f_{2}}{\partial x} & \dfrac{\partial f_{2}}{\partial y} \end{bmatrix} = \begin{bmatrix} 2xy & x^2 \\ 5 & \cos y \end{bmatrix}\]and the Jacobian determinant is equal to
\[\operatorname{det}\left(\mathbf{J}_{\mathbf{f}}(x, y)\right)=2 x y \cos y-5 x^{2}\]Example 2: Polar-Cartesian Transformation
The transformation from polar coordinates \((r, \varphi)\) to Cartesian coordinates \((x, y)\), is given by the function \(\mathbf{F} : \mathbb{R}^{+} \times[0,2 \pi) \rightarrow \mathbb{R}^{2}\) with components:
\[x = r \cos \varphi\] \[y = r \sin \varphi\]Therefore, the Jacobian matrix of \(f\) is given by:
\[\mathbf{J}_{f}(r, \varphi) = \begin{bmatrix} \dfrac{\partial x}{\partial r} & \dfrac{\partial x}{\partial \varphi} \\ \dfrac{\partial y}{\partial r} & \dfrac{\partial y}{\partial \varphi} \end{bmatrix} = \begin{bmatrix} \cos \varphi & -r \sin \varphi \\ \sin \varphi & r \cos \varphi \end{bmatrix}\]and the Jacobian determinant is equal to \(r\). It is used to transform integrals between the two coordinate systems:
\[\iint_{\mathbf{F}(A)} f(x, y) d x d y=\iint_{A} f(r \cos \varphi, r \operatorname{sir} \varphi) r d r d \varphi\]Example 3: Spherical-Cartesian Transformation
The transformation from spherical coordinates \((r, \theta, \varphi)\) to Cartesian coordinates \((x, y, z)\) is given by the function \(\mathbf{F} : \mathbb{R}^{+} \times[0, \pi] \times[0,2 \pi) \rightarrow \mathrm{R}^{3}\) with components:
\(x=r \sin \theta \cos \varphi\)
\(y=r \sin \theta \sin \varphi\)
\(z=r \cos \theta\)
The Jacobian matrix for this coordinate change is given by:
\[\mathbf{J}_{f}(r, \theta, \varphi) = \begin{bmatrix} \dfrac{\partial x}{\partial r} & \dfrac{\partial x}{\partial \theta} & \dfrac{\partial x}{\partial \varphi} \\ \dfrac{\partial y}{\partial r} & \dfrac{\partial y}{\partial \theta} & \dfrac{\partial y}{\partial \varphi} \\ \dfrac{\partial z}{\partial r} & \dfrac{\partial z}{\partial \theta} & \dfrac{\partial z}{\partial \varphi} \end{bmatrix}\] \[=\begin{bmatrix} \sin \theta \cos \varphi & r \cos \theta \cos \phi & -r \sin \theta \sin \varphi \\ \sin \theta \sin \varphi & r \cos \theta \sin \varphi & r \sin \theta \cos \varphi \\ \cos \theta & - r \sin \theta & 0 \end{bmatrix}\]and the Jacobian determinant is equal to \(r^{2} \sin \theta\).
As an example, since \(dV=dxdydz\), this determinant implies that the differential volume element \(d V=r^{2} \sin \theta d r d \theta d \varphi\).
PYQs
Limits
1) Find \(\lim_{(x, y) \rightarrow(0,0)} \dfrac{x^{2} y}{x^{3}+y^{3}}\) 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 \(f(x, y)=\left\{\begin{array}{ll}{\dfrac{2 x^{4}y-5 x^{3} y^{2}+y^{5}}{\left(x^{2}+y^{2}\right)^{2}},} & {(x, y ) \neq(0,0)} \\ {0} & {, (x, y )=(0,0)}\end{array}\right.\)
Find a \(\delta>0\) such that \(\vert f(x, y)-f(0,0) \vert< 0.01\) whenever \(\sqrt{x^{2}+y^{2}}<\delta\).
[2016, 15M]
Remarks: Original question has printing mistake
3) Examine the continuity for the function \(f(x, y)=\left\{\begin{array}{cc}{\dfrac{x^{2}-x \sqrt{y}}{x^{2}+y}} & {,(x, y) \neq(0,0)} \\ {0} & {,(x, y)=(0,0)}\end{array}\right.\)
[2015, 12M]
4) Define a function \(f\) of two real variables in the plane by: \(f(x, y)=\left\{\begin{array}{c}{ \dfrac{x^{3} \cos \dfrac{1}{y}+y^{3} \cos \dfrac{1}{x}}{x^{2}+y^{2}}, \text { for } x, y \neq 0 } \\ {0, \text { otherwise }}\end{array}\right.\)
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.
\[f(x, y)=\left\{\begin{array}{l}{0 \text { if } x y=0} \\ {1 \text { if } x y \neq 0}\end{array}\right.\][2005, 12M]
Partial Derivatives
1) If
\[u=sin^{-1} \sqrt{\dfrac{x^{1/3}+y^{1/3}}{x^{1/2}+y^{1/2}}},\]then show that \(\sin^2 u\) is homogeneous function of \(x\) and \(y\) of degree \(-\dfrac{1}{6}\). Hence show that
\(x^2\dfrac{\partial^2 u}{\partial x^2}+2xy\dfrac{\partial^2 u}{\partial x\partial y}+y^2\dfrac{\partial^2 u}{\partial y^2}\)=\(\dfrac{\tan u}{12}(\dfrac{13}{12}+\dfrac{\tan^2 u}{12})\)
[2018, 12M]
2) Let
\(f(x,y) = xy^2\), if \(y>0\) and \(f(x,y) = -xy^2\), if \(y \leq 0\)
Determine which of \(\dfrac{\partial f}{\partial x}(0,1)\) and \(\dfrac{\partial f}{\partial y}(0,1)\) exists and which does not exist.
[2018, 12M]
3) If \(f(x, y)=\left\{\begin{array}{ll}{\dfrac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}},(x, y) \neq(0,0)} \\ {0, (x, y)=(0,0)}\end{array}\right.\)
Calculate \(\dfrac{\partial^{2} f}{\partial x \partial y}\) and \(\dfrac{\partial^{2} f}{\partial y \partial x}\) at (0,0).
[2017, 15M]
4) Compute \(f_{x y}(0,0)\) and \(f_{y x}(0,0)\) for the function
\(f(x, y)=\left\{\begin{array}{l}{\dfrac{x y^{3}}{x+y^{2}},(x, y) \neq(0,0)} \\ {0 \quad,(x, y)=(0,0)}\end{array}\right.\)
Also discuss the continuity of \(f_{x y}\) and \(f_{yx}\) 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) \(x \dfrac{\partial f}{\partial x}+y \dfrac{\partial f}{\partial y}=n f\)
ii) \(x^{2} \dfrac{\partial^{2} f}{\partial x^{2}}+2 x y \dfrac{\partial^{2} f}{\partial x \partial y}+y^{2} \dfrac{\partial^{2} f}{\partial y^{2}}=n(n-1) f\).
[2010, 10M]
6) Let \(f: R^2 \rightarrow R\) be defined as:
\[f(x, y)=\left\{\begin{array}{ll}{\dfrac{x y}{\sqrt{x^{2}+y^{2}}},(x, y) \neq(0,0)} \\ {0, (x, y)=(0,0)}\end{array}\right.\]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 \(\theta\) of the point \(P(x, y)\). Express your estimates as percentage changes of the value that \(r\) and \(\theta\) have at the point (3,4).
[2009, 20M]
8) Prove that if \(z=\phi(y+a x)+\psi(y-a x)\), then \(a^{2} \dfrac{\partial^{2} z}{\partial y^{2}}-\dfrac{\partial^{2} z}{\partial x^{2}}=0\) for any twice differentiable \(\phi\) and \(\psi\). Here, \(a\) is a constant.
[2007, 15M]
9) If \(z=x f\left(\dfrac{y}{x}\right)+g\left(\dfrac{y}{x}\right)\), show that \(x^{2} \dfrac{\partial^{2} z}{\partial x^{2}}+2xy \dfrac{\partial^{2} z}{\partial x \partial y}+y^{2} \dfrac{\partial^{2} z}{\partial y^{2}}=0\)
[2006, 15M]
10) Let \(R^{2} \rightarrow R\) be defined as \(f(x, y)=\dfrac{x y}{\sqrt{\left(x^{2}+y^{2}\right)}},(x, y) \neq(0,0)=0\), 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: \(f(x, y)=\left\{\begin{array}{ll}{\dfrac{x y}{x^{2}+y^{2}},(x, y) \neq(0,0)} \\ {0}, (x, y)=(0,0)\end{array}\right.\) 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 \(x^4-5x^2+4\) 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 \(\left(x^{2}+y^{2}+z^{2}\right)=a^{2} x^{2}+b^{2} y^{2}+c^{2} z^{2}\) 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 \(\dfrac{x^{2}}{4}+\dfrac{y^{2}}{5}+\dfrac{z^{2}}{25}=1\) 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 \(\dfrac{x^{2}}{4}+\dfrac{y^{2}}{9}=1\)
[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 \(f'(x)=\dfrac{1}{1+x^2}\) and \(f(0)=0\), then
\[f(x)+f(y)=f\left(\dfrac{x+y}{1-xy}\right)\][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 \(\dfrac{\partial(x, y, z)}{\partial(u, v, w)}\)
[2005, 15M]
4) Let the roots of the below equation in \(\lambda\) be \(u, v, w\). \((\lambda-x)^{3}+(\lambda-y)^{3}+(\lambda-z)^{3}=0\)
Prove that: \(\dfrac{\partial(u, v, w)}{\partial(x, y, z)}=-2 \dfrac{(y-z)(z-x)(x-y)}{(u-v)(v-w)(w-u)}\)
[2004, 15M]
5) If the roots of the equation \((\lambda-u)^{3}+(\lambda-v)^{3}+(\lambda-w)^{3}=0\) in \(\lambda\) are \(x, y, z\), show that:
\[\dfrac{\partial(x, y, z)}{\partial(u, v, w)}=-\dfrac{2(u-v)(v-w)(w-u)}{(x-y)(y-z)(z-x)}\][2002, 15M]