Answer to Question #237863 in Mechanics | Relativity for Annie

Question #237863

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


1
Expert's answer
2021-09-17T09:23:49-0400

Q6.

grad f1=2xy3z4i+3x2y2z4j+4x2y3z3k,\text{grad }f_1=2xy^3z^4\vec i+3x^2y^2z^4\vec j+4x^2y^3z^3\vec k,

gradf2=exsinylnzi+excosylnzj+exsinyzk,\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)

H0=(x0,y0)=(2,3),H_0=(x_0,y_0)=(-2,3),

b)

h0=hH0=720,h_0=h|_{H_0}=720,

c)

α=arctanhyhx(1,1)=arctan2x8y+282y6x+18(1,1)=arctan2214=57.5°,\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)

divV=2x(y1),\text{div}\vec V=-2x(y-1),

curlV=(2xz,2yz,3y2),\text{curl}\vec V=(-2xz,2yz,3y^2),

b)

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

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

c)

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

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


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment