Comments on: Vector Calculus: Understanding the Gradient http://betterexplained.com/articles/vector-calculus-understanding-the-gradient/ Learn Right, Not Rote. Wed, 16 May 2012 12:30:32 +0000 hourly 1 http://wordpress.org/?v= By: kalid http://betterexplained.com/articles/vector-calculus-understanding-the-gradient/#comment-57254 kalid Sun, 26 Feb 2012 09:07:35 +0000 http://betterexplained.com/articles/vector-calculus-understanding-the-gradient/#comment-57254 @Deniz: Thanks! And you're welcome :). @Deniz: Thanks! And you’re welcome :) .

]]> By: kalid http://betterexplained.com/articles/vector-calculus-understanding-the-gradient/#comment-57227 kalid Sun, 26 Feb 2012 07:13:36 +0000 http://betterexplained.com/articles/vector-calculus-understanding-the-gradient/#comment-57227 @Chico: Awesome, thanks for sharing! I like that a lot -- lining up with the gradient (out of all possible directional derivatives) will give you the best return (cosine = 1). That clicks for me. Glad you enjoyed the microwave intuition, I love searching for little analogies. @Chico: Awesome, thanks for sharing! I like that a lot — lining up with the gradient (out of all possible directional derivatives) will give you the best return (cosine = 1). That clicks for me.

Glad you enjoyed the microwave intuition, I love searching for little analogies.

]]> By: Deniz http://betterexplained.com/articles/vector-calculus-understanding-the-gradient/#comment-56217 Deniz Tue, 21 Feb 2012 21:06:33 +0000 http://betterexplained.com/articles/vector-calculus-understanding-the-gradient/#comment-56217 I already knew this but you gave me a better intuition of it and I like your style of writing! Thank you! :) I already knew this but you gave me a better intuition of it and I like your style of writing! Thank you! :)

]]> By: Chico http://betterexplained.com/articles/vector-calculus-understanding-the-gradient/#comment-54805 Chico Fri, 17 Feb 2012 03:54:32 +0000 http://betterexplained.com/articles/vector-calculus-understanding-the-gradient/#comment-54805 Consider the directional derivative, f_u. f_u = f_x u_1 + f_y u_2 (it takes some effort to see this definition of f_u) =grad(f) dot u (u is a unit vector) =|grad(f)| cos@ (@ is the angle between grad(f) and u) Thus, it is clear that the directional derivative, f_u, is maxed when cos@=1. It follows that @=0 and the directional derivative, f_u, is attained when u is in the direction of the gradient. Therefore, the gradient does indeed give the direction of greatest increase. Note that f_u is minimized when cos@=-1. Thus, @=pi, and u is in the opposite direction of the gradient. QED ps I am a nerdy math professor who likes demonstrating mathematical prowess. Thanks for the microwave intuition builder. My students are going to like that. Consider the directional derivative, f_u.
f_u = f_x u_1 + f_y u_2 (it takes some effort to see this definition of f_u)
=grad(f) dot u (u is a unit vector)
=|grad(f)| cos@ (@ is the angle between grad(f) and u)
Thus, it is clear that the directional derivative, f_u, is maxed when cos@=1.
It follows that @=0 and the directional derivative, f_u, is attained when u is in the direction of the gradient. Therefore, the gradient does indeed give the direction of greatest increase.

Note that f_u is minimized when cos@=-1. Thus, @=pi, and u is in the opposite direction of the gradient. QED

ps
I am a nerdy math professor who likes demonstrating mathematical prowess. Thanks for the microwave intuition builder. My students are going to like that.

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