Vector Calculus: Understanding the Gradient

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21

Hi Caitlyn, you’re welcome.

22

Thanks! The sadistic microwave example helped a lot.

23

Awesome, glad it was useful :).

24

Hello Kalid,
Did not read your reply for some
time. Am sorry you do not agree. :slight_smile:

Let me give you an example:
Suppose we are dealing with pressure
and height in a certain ‘cubic’
area. Suppose that the middle of the
cube height is 0 meters. Also suppose
that we have a whirlpool generated in the
cube such that the pressure rate increases
as we go below the middle of the cube.
Anything below is negative height and anything above
is positive height. Now, as one rises
higher in the cube, the pressure decreases.
If we find the gradient, then according to
your definition (and many others’), then
the gradient vector for the rate of greatest
increase will point below the middle of the
cube, not above. But above the middle we
find the greatest ‘decrease’ in rate of pressure.

In this example, greatest increase points
downwards and greatest decrease upwards.

It would probably be better to define
gradient as a vector that points in a
direction of greatest increase or decrease.
It’s additive inverse will point in the
diretion of greatest decrease or increase
respectively. For most physical phenomena,
your definition would generally be true.
But what happens when you have an anomaly?
Make sense?

25

I do not believe I have the best answer to this question but like yourself, I am a believer in trying to find the best possible explanation. Once again, I like your website. Keep up the good work Kalid!

26

Okay, I think I have the best answer. If f is a real-valued function, then del(f) or gradient of f points to the greatest increase, whereas -del(f) points t0 the greatest decrease.

For once planet math has some decent information on this since I last checked:

http://planetmath.org/encyclopedia/Gradient.html

I do not endorse everything Planet Math publishes but this particular information appears to be correct. In any event, it clears up the previous confusion I think.

27

Hi John, thanks for the comment! Yes, that’s an important distinction to make: the positive gradient is the greatest increase, and the negative gradient is the greatest decrease. Thanks for helping clarify :).

28

Thank you!

29

This actually makes sense to me. Thanks!

30

@Jared, Bigmouth: Cool, glad it was helpful!

31

did not grasp the idea

32

Be more specific. The gradient is the direction to move that gives you the biggest increase.

33

It helps me a lot. But I have some doubt still now.Is it the same concept for gradient of each vertex in a triangle mesh?

Thanks so much.

34

Kalid

Thanks for the great explanations! I thought I was math-retarded for some time; however your writings actually make sense to me!

Take care!

Johnny T

35

@Shaheen: Thanks, glad you enjoyed it. I’m not sure I understand the question: in a triangle mesh, you could measure the gradient at each vertex to find the “best” direction to move. Again, not sure if this is your question.

@Johnny T: Thank you for the comment! Yes, when a subject seems difficult (as vector calculus was for me) sometimes it’s just because the explanation wasn’t clicking properly. Thanks for dropping by.

36

well done,excellent explaination with solid examples

37

Thanks Wali, glad you enjoyed it.

38

[…] Div, Grad, Flux and Curl (if you already know vector calculus) […]

39

thanks
but i have some doubts.how the differentaion gives the maximum space rate of change. as per my understandings differentiation only is difference between two point in the region say p1 and p2.can u clarify

40

Thanks a lot for explaining the concept.