Question

Q6. Find the gradient of the following functions

f(x, y, z) = x2y3z4.

f(x, y, z) = exsin(y) ln(z).

Q7. The height of certain hill (in feet) is given by h(x,y)=10(2xy−3x2 −4y2 −18x+28y+12)

Where y is the distance in north, x is the distance in east.

a. Where is the top of the hill located?

b. How high is the hill?

c. How steep is the slope (in feet per mile) at a point 1-mile north and 1- mile east, in what

direction is the slope steepest?

Q8. Calculate the divergence and Curl, of the following functions

a. V= x2 i+3xy^2 j -2xyz k

b. V=xyi+2yz j+3zx k

c. V= y2i +(2xy+z2)j+ 2yz k , where i , j , k are unit vectors along x, y and z axis

Answer 1

Q6.

$\text{grad }f_1=2xy^3z^4\vec i+3x^2y^2z^4\vec j+4x^2y^3z^3\vec k,$

$\text{grad}f_2=e^x\sin y\ln z\vec i+e^x\cos y\ln z\vec j+\frac{e^x\sin y}z\vec k,$

Q7.

a)

$H_0=(x_0,y_0)=(-2,3),$

b)

$h_0=h|_{H_0}=720,$

c)

$\alpha=-\arctan {\frac{h'_y}{h'_x}}|_{(1,1)}=-\arctan{\frac{2x-8y+28}{2y-6x+18}}|_{(1,1)}=-\arctan \frac{22}{14}=-57.5°,$

Q8.

a)

$\text{div}\vec V=-2x(y-1),$

$\text{curl}\vec V=(-2xz,2yz,3y^2),$

b)

$\text{div}\vec V=3x+y+2z,$

$\text{curl}\vec V=(-2y,-3z,-x),$

c)

$\text{div}\vec V=2(x+y),$

$\text{curl}\vec V=(0,0,0).$