Directional derivative question

[question dude]

I've done the first part, but I'm stuck on the second paragraph of the question. Maybe I'm being stupid, I don't even understand exactly what is meant by, 'the level curve'.

I also don't quite understand the whole concept of directional derivative. When it says, 'the gradient in the direction making an angle A with the x-axis, how should I think of this? what does the gradient mean in this context?

because when its just a simple curve/line on an xy axis, I know what the 'gradient' means, it is literally how much 'y' changes per unit x along that curve/line. But when I'm told about a 'gradient in a direction', I'm confused.

 

[tiny-tim]

hey, question dude!

… what is meant by, 'the level curve'.

think of the 3D graph, z = f(x,y)

you can make a 2D contour map showing the lines of equal height

those contours are the level curves

I also don't quite understand the whole concept of directional derivative. When it says, 'the gradient in the direction making an angle A with the x-axis, how should I think of this? what does the gradient mean in this context?

the directional derivative in the direction (cosθ,sinθ) is df(kcosθ,ksinθ)/dk

it's the rate at which f increases if you go along the line y/x = tanθ

 

[Ray Vickson]

I've done the first part, but I'm stuck on the second paragraph of the question. Maybe I'm being stupid, I don't even understand exactly what is meant by, 'the level curve'.

I also don't quite understand the whole concept of directional derivative. When it says, 'the gradient in the direction making an angle A with the x-axis, how should I think of this? what does the gradient mean in this context?

because when its just a simple curve/line on an xy axis, I know what the 'gradient' means, it is literally how much 'y' changes per unit x along that curve/line. But when I'm told about a 'gradient in a direction', I'm confused.

You are supposed to show your work, confused or not (those are PF rules). I don't see how you can have done the first part (which involves directional derivatives) but then claim you do not understand directional derivatives. I would need to see your work in order to grasp what is going on.

 

[question dude]

hey, question dude!


think of the 3D graph, z = f(x,y)

you can make a 2D contour map showing the lines of equal height

those contours are the level curves


the directional derivative in the direction (cosθ,sinθ) is df(kcosθ,ksinθ)/dk

it's the rate at which f increases if you go along the line y/x = tanθ

Thanks a lot! I understand what's going on now. Before I was just plugging in the numbers without knowing the concepts properly.

 

[question dude]

You are supposed to show your work, confused or not (those are PF rules). I don't see how you can have done the first part (which involves directional derivatives) but then claim you do not understand directional derivatives. I would need to see your work in order to grasp what is going on.

Sorry about that, what I meant to get across, was that I understood the method in a sort of algorithmic way, but didn't understood the reason behind steps due to not understanding some basic stuff like 'level curve'. Anyway its sorted now.