Vector calculus - How to use the gradient?

[alexvenk]

I have done part A (i think) really not sure where to begin with the rest of the parts, would appreciate a tip in the right direction, its revision for my first year physics exams in a few weeks.

Consider the funtion T in the plane (x,y), given by T=ln(x^2 + y^2)

at point 1,2

a) in which direction is most rapid increase in T

I did Grad(T) to get a vector which i think is in the direction of most rapid increase (2/5,4/5)

b) what distance in this direction gives an inrease of .2 in T

c) what distance in direction i + j gives and increase of .12 in T

d) in what directions will T be stationary.

I don't want solutions, just how to go about solving the problems

 

[alexvenk]

I have done part A (i think) really not sure where to begin with the rest of the parts, would appreciate a tip in the right direction, its revision for my first year physics exams in a few weeks.

Consider the funtion T in the plane (x,y), given by T=ln(x^2 + y^2)

at point 1,2

a) in which direction is most rapid increase in T

I did Grad(T) to get a vector which i think is in the direction of most rapid increase (2/5,4/5)

b) what distance in this direction gives an inrease of .2 in T

c) what distance in direction i + j gives and increase of .12 in T

d) in what directions will T be stationary.

I don't want solutions, just how to go about solving the problems

 

[Ray Vickson]

I have done part A (i think) really not sure where to begin with the rest of the parts, would appreciate a tip in the right direction, its revision for my first year physics exams in a few weeks.

Consider the funtion T in the plane (x,y), given by T=ln(x^2 + y^2)

at point 1,2

a) in which direction is most rapid increase in T

I did Grad(T) to get a vector which i think is in the direction of most rapid increase (2/5,4/5)

b) what distance in this direction gives an inrease of .2 in T

c) what distance in direction i + j gives and increase of .12 in T

d) in what directions will T be stationary.

I don't want solutions, just how to go about solving the problems

For (b): if you go along direction (2/5,4/5) from the point (1,2) you are looking at points of the form $x = x(t) = 1 + (2/5)t, \:y = y(t) = 2 + (4/5)t$, where $t > 0$ is a scalar.

 

[alexvenk]

(DelT = delta T (change in T))

Turns out the best way to do it for those who are interested is you use DelT = GradT . r, to get the largest change in t (highest delT) r and GradT must be in the same direction. To work out how far in a certain direcction it changes by a certain amount, set delT to whatever you want the change to be (.2 for b) then set r to be a vecctor with magnitude a and direction the same as the direction it was in a, then simply solve for a. Do the same for part c, and finally for part d, set delT to 0 so GradT must be perpendicular to r, which is pretty easy to do by inspection.

Thanks for the reply.

 

[theodoros.mihos]

For simplicity, direction of (2/5,4/5) is the same of (1,2).