Vector Calculus

The maximum value of the directional derivative takes place in the direction of  $\nabla\Phi$  and its magnitude is

•  $\left|\nabla\Phi\right|$  

•  $\left|\nabla^2\Phi\right|$  

•  $\left|\sqrt{\nabla\Phi}\right|$  

•  $\nabla\Phi$  

If  $\phi$  (x,y,z) = $xy^2+yz^3\$  then grad $\phi$  is

•  $x^2i+2xyj+3z^2k$  

•  $y^2i-\left(2xy+z^{3\ }\right)j+3yz^2k$  

•  $y^2i+\left(2xy+z^3\right)j+3yz^2k$  

•  $y^2i+z^2j-x^2k$  

The unit normal to the surface  $x^3-xyz+z^3=1$  at (1,1,0) is

•  $\frac{1}{3}\left(2i-j+2k\right)$  

•  $\frac{\left(3i-k\right)}{\sqrt{10}}$  

•  $\frac{1}{\sqrt{10}}\left(3i+j-k\right)$  

•  $\frac{1}{3}\left(3i-k\right)$  

Determine the validity of the statements.

a) The divergence of a scalar valued function is vector valued function.

b) The curl of a vector valued function is vector valued function.

c) A vector f is solenoidal if div f=o.

d) A vector is called harmonic if curl f=o vector.

• a and b are true

• only d is true

• b and c are true

• only c is true

The vector  $f=\left(x^2-yz\right)i+\left(y-zx\right)j+\left(z^2-xy\right)k$  is 

• irrotational

• solenoidal

• linear

• harmonic

If  $r=xi+yj+zk$  then curl r and div r is

• 0,3

• 3,0

• 1,1

• 1,0

If  $\phi\left(x,y,z\right)$  is any solution of Laplace's equation then  $\nabla\phi$  is

• Irrotational

• Solenoidal

• harmonic

• Irrotational and solenoidal

div curl f= o vector and curl grad  $\phi$  =0. State true or false. 

• True

• False

The directional derivative of  $\phi=xy^2+yz^3$  at the point (2,-1,1) is

•  $i-3j-3k$  

•  $i+3j-3k$  

•  $i+3j+3k$  

•  $-i+3j-3k$  

The volume of the parallelepiped formed by the coterminous edges a,b,c is represented by

• $a\times\left(b\times c\right)$

• $a.\left(b\times c\right)$

• $a\times\left(b.c\right)$

• $a.\left(b.c\right)$