UNIT II- FUNCTIONS OF SEVERAL VARIABLES

 If u and v are functions of x and y then u and v are said to be functionally related if ,

•  $J=\frac{\partial\left(u\ ,v\right)}{\partial\left(x\ ,\ y\right)}\ =\ 1$  

•  $J\ =\ \frac{\partial\left(u\ ,\ v\right)}{\partial\left(x\ ,\ y\right)}\ =\ 0$  

•  $J\ =\ \frac{\partial\left(u\ ,\ v\right)}{\partial\left(x\ ,\ y\right)}\ =\ \infty$  

• None of these

If u is a homogeneous function of degree ' n' in x and y , then

• $x\frac{\partial u}{\partial x}\ +\ y\ \ \frac{\partial u}{\partial y}\ =\ n$

• $x\frac{\partial u}{\partial x}\ +\ y\ \ \frac{\partial u}{\partial x}\ =\ nu$

• $x\frac{\partial u}{\partial x}\ +\ y\ \ \frac{\partial u}{\partial y}\ =\ nu$

• $x\frac{\partial u}{\partial x}\ +\ y\ \ \frac{\partial u}{\partial y}\ =\ u$

If u , v are functions of r, s and r , s are functions of x , y then

• $\frac{\partial\left(u\ ,\ v\right)}{\partial\left(x\ ,\ y\right)}\ =\ \frac{\partial\left(u\ ,\ v\right)}{\partial\left(r\ ,\ s\right)}\ \times\frac{\partial\left(u\ ,\ v\right)}{\partial\left(x\ ,\ y\right)}$

• $\frac{\partial\left(u\ ,\ v\right)}{\partial\left(x\ ,\ y\right)}\ =\ \frac{\partial\left(u\ ,\ v\right)}{\partial\left(r\ ,\ s\right)}\ \times\frac{\partial\left(r\ ,\ s\right)}{\partial\left(x\ ,\ y\right)}$

• $\frac{\partial\left(u\ ,\ v\right)}{\partial\left(x\ ,\ y\right)}\ =\ \frac{\partial\left(x\ ,\ v\right)}{\partial\left(r\ ,\ s\right)}\ \times\frac{\partial\left(r\ ,\ s\right)}{\partial\left(x\ ,\ y\right)}$

• $\frac{\partial\left(u\ ,\ v\right)}{\partial\left(x\ ,\ y\right)}\ =\ \frac{\partial\left(u\ ,\ v\right)}{\partial\left(r\ ,\ s\right)}\ \times\frac{\partial\left(r\ ,\ s\right)}{\partial\left(u\ ,\ v\right)}$

For function f(x,y) of two variables condition for minimum at certain point is

• $rt\ -\ s^2\ >\ 0\ and\ r\ >\ 0$

• $rt\ -\ s^2\ <\ 0\ and\ r\ <\ 0$

• $rt\ -\ s^2\ >\ 0\ and\ r\ <\ 0$

• $rt\ -\ s^2\ =\ 0\ and\ r\ >\ 0$

 $x\ =\ r\cos\theta\ ,\ y\ =\ r\sin\theta\ then\ \frac{\partial\left(r,\ \theta\right)}{\partial\left(x\ ,\ y\right)}\ =\ ?$  

If

• 0

• 1

•  $\frac{1}{r}$  

•  $r$  

If  $f\left(x\ ,\ y\ ,z\right)$  be a function of  $x\ ,\ y\ ,\ z$   which is to be examined for maximum or minimum value.Let the variables  $x\ ,\ y\ ,\ z$   be connected by the relation  $\phi\left(x\ ,\ y\ ,z\right)\ =\ 0\$  then by Lagrange's method of multiplier....

•  $\frac{\partial f}{\partial x}\ +\lambda\ \frac{\partial\phi}{\partial x}=1\ ,\frac{\partial f}{\partial y}\ +\lambda\ \frac{\partial\phi}{\partial y}\ =1\ ,\frac{\partial f}{\partial z}\ +\lambda\ \frac{\partial\phi}{\partial z}\ =\ 1$  

•  $\frac{\partial f}{\partial x}\ +\lambda\ \frac{\partial\phi}{\partial x}\ =0\ ,\frac{\partial f}{\partial y}+\lambda\ \frac{\partial\phi}{\partial y}=0\ ,\frac{\partial f}{\partial z}\ +\lambda\ \frac{\partial\phi}{\partial z}\ =0$  

•  $\frac{\partial f}{\partial x}+\lambda\ \frac{\partial\phi}{\partial x}\ =1\ ,\frac{\partial f}{\partial y}+\lambda\ \frac{\partial\phi}{\partial y}\ =0\ ,\frac{\partial f}{\partial z}\ +\lambda\ \frac{\partial\phi}{\partial z}\ =1$  

• None of these

 $If\ u\ =\ \frac{x\ +\ y}{x\ -\ y}\ and\ v\ =\ \frac{x\ -\ y}{\left(x\ -\ y\right)^2}\$  then the relation is ....

•  $u^2\ =\ 1+4v$  

•  $u\ =\ 1+\ 4\ v^2$  

•  $u^2\ =\ 4v$  

•  $u^2\ =\ u+4v$  

 $If\ x\ =\ \sqrt{vw}\ ,\ y\ =\ \sqrt{wu}\ ,\ z\ =\sqrt{uv}\ then\frac{\partial\left(x\ ,\ y\ ,\ z\right)}{\partial\left(u\ ,\ v\ ,\ w\right)}\ =\ ?$  

• 4

• 1

•  $\frac{1}{4}$  

•  $\frac{1}{8}$  

 $If\ u\ =\ x^y\ then\ \frac{\partial u}{\partial y}\ is\ equal\ to\$  

• 0

•  $y\ x^{\left(y\ -\ 1\right)}$  

•  $x^y\ \log x$  

• None

 $If\ u\ =\ \tan^{-1}\left(\frac{x^{3\ }+y^{3\ }\ }{x\ +\ y}\right)\ then\ ...\ x\frac{\partial u}{\partial x}\ +y\ \frac{\partial u}{\partial y}\ =\ ?$  

•  $\sin\ 2u$  

•  $\cos\ 2u$  

•  $2\ \tan\ u$  

•  $2\ \sin\ u$