Vector Calculus: Understanding the Gradient

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41

i was having so much trouble understanding this and now its all clear thank you so much!

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@lon, sophie: Thanks, glad you enjoyed it!

43

Jesus. This was a lot better explained than in my text book and by my professor. I thought we were using the gradient as the normal vector but I really doubted that it could be that.

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@Ryan: Thanks! I struggled with this concept for a while also.

45

thanks ! this explanation made me clear how to find the direction of smallest change.It is just the 90 degree rotation of gradiant(the direction of largest change).

46

Thanks very much for your effort

47

Um – in your microwave example, aren’t you pushing the doughboy out the back of the microwave? (Just wanted to understand the concept). I love these essays, btw, keep them coming!

48

I loved the microwave analogy.also thanks for clarifying the upsidedown delta now everything makes more sense

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stil im confused between scalar field and vector field…

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[…] Vector Calculus: Gradient, Flux, Divergence, Curl & Circulation […]

51

how can such a mathematical expression denote the max change? pls i didnt understand the relation of this with mathematics. pls reply sir.

52

thank you soo much!!

its a big help for our project…

Can we have your number?hehe

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@Rahul: A scalar field returns a single value (x), but a vector field returns multiple values (x,y,z). Usually the multiple values (x,y,z) are taken as a “direction” to follow.

@aradhita: Hi, that’s a question I need to get into in a later post.

@nat2_bam2: Thanks!

54

Hi kalid! i read your explanation. oh this is very helpful! by the way can you give an example on how to apply this on a situation of the classic “mountain and mountain climber” problem? hope you will reply. thanks again your explanations were clear

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@Migs: Great question. The classic “mountain climber” problem is when the vector field gives the height of the mountain (z) at a certain position (x,y), so z = f(x,y).

The gradient at any position x,y will give you the direction of the greatest increase in z. That is, the gradient will point in the “most uphill”. Following the gradient will give you the shortest path the the top of the mountain (technically, the top of the nearest local maximum). How this helps!

56

beautiful…well said

57

thanks a lot for the wonderful explanation!!!

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@akansha: You’re welcome!

59

Very nice! Keep up. Thanks a lot

60

Very nice article!!
Hope to see how to find the maximum of a constrained function soon!!
Thanks a lot!!