Gradient of a function points in the direction of the greatest rate of increase of the function.
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Somewhere I saw that the gradient of a function points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. I don't understand it at all. Can somebody help me to understand it? Please give some example?
calculus multivariable-calculus
asked Sep 6 at 5:29
Infinity
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up vote
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Somewhere I saw that the gradient of a function points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. I don't understand it at all. Can somebody help me to understand it? Please give some example?
calculus multivariable-calculus
asked Sep 6 at 5:29
Infinity
518213
Possible duplicate of math.stackexchange.com/questions/223252/…
– saulspatz
Sep 6 at 5:44
You might also have a look at math.stackexchange.com/q/1912660/265466.
– amd
Sep 6 at 6:39
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Somewhere I saw that the gradient of a function points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. I don't understand it at all. Can somebody help me to understand it? Please give some example?
calculus multivariable-calculus
asked Sep 6 at 5:29
Infinity
518213
Somewhere I saw that the gradient of a function points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. I don't understand it at all. Can somebody help me to understand it? Please give some example?
calculus multivariable-calculus
calculus multivariable-calculus
asked Sep 6 at 5:29
Infinity
518213
asked Sep 6 at 5:29
Infinity
518213
asked Sep 6 at 5:29
Infinity
518213
asked Sep 6 at 5:29
Infinity
518213
asked Sep 6 at 5:29
Infinity
518213
518213
Possible duplicate of math.stackexchange.com/questions/223252/…
– saulspatz
Sep 6 at 5:44
You might also have a look at math.stackexchange.com/q/1912660/265466.
– amd
Sep 6 at 6:39
add a comment |Â
Possible duplicate of math.stackexchange.com/questions/223252/…
– saulspatz
Sep 6 at 5:44
You might also have a look at math.stackexchange.com/q/1912660/265466.
– amd
Sep 6 at 6:39
Possible duplicate of math.stackexchange.com/questions/223252/…
– saulspatz
Sep 6 at 5:44
Possible duplicate of math.stackexchange.com/questions/223252/…
– saulspatz
Sep 6 at 5:44
You might also have a look at math.stackexchange.com/q/1912660/265466.
– amd
Sep 6 at 6:39
You might also have a look at math.stackexchange.com/q/1912660/265466.
– amd
Sep 6 at 6:39
add a comment |Â
1 Answer
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The link I gave has very good answers based on the directional derivative that prove the gradient points in the direction of steepest ascent, using directional derivatives. What has made this result intuitive for me is another result: the gradient is perpendicular to the level curve. Imagine standing on a hillside, and visualize the curve passing through all points at the same altitude as the point you're standing at. It just seems clear to me from experience, that the steepest climb is perpendicular to that curve (the direction of the gradient), and that if if you were to set a ball on the ground and let it go, it would roll downhill perpendicular to that curve (in the direction of the negative of the gradient.)
The link I gave proves the theorem for a function of $3$ variables, but the same proof works in any number of dimensions of course.
answered Sep 6 at 6:07
saulspatz
11.6k21324
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
The link I gave has very good answers based on the directional derivative that prove the gradient points in the direction of steepest ascent, using directional derivatives. What has made this result intuitive for me is another result: the gradient is perpendicular to the level curve. Imagine standing on a hillside, and visualize the curve passing through all points at the same altitude as the point you're standing at. It just seems clear to me from experience, that the steepest climb is perpendicular to that curve (the direction of the gradient), and that if if you were to set a ball on the ground and let it go, it would roll downhill perpendicular to that curve (in the direction of the negative of the gradient.)
The link I gave proves the theorem for a function of $3$ variables, but the same proof works in any number of dimensions of course.
answered Sep 6 at 6:07
saulspatz
11.6k21324
add a comment |Â
up vote
0
down vote
The link I gave has very good answers based on the directional derivative that prove the gradient points in the direction of steepest ascent, using directional derivatives. What has made this result intuitive for me is another result: the gradient is perpendicular to the level curve. Imagine standing on a hillside, and visualize the curve passing through all points at the same altitude as the point you're standing at. It just seems clear to me from experience, that the steepest climb is perpendicular to that curve (the direction of the gradient), and that if if you were to set a ball on the ground and let it go, it would roll downhill perpendicular to that curve (in the direction of the negative of the gradient.)
The link I gave proves the theorem for a function of $3$ variables, but the same proof works in any number of dimensions of course.
answered Sep 6 at 6:07
saulspatz
11.6k21324
add a comment |Â
up vote
0
down vote
up vote
0
down vote
The link I gave has very good answers based on the directional derivative that prove the gradient points in the direction of steepest ascent, using directional derivatives. What has made this result intuitive for me is another result: the gradient is perpendicular to the level curve. Imagine standing on a hillside, and visualize the curve passing through all points at the same altitude as the point you're standing at. It just seems clear to me from experience, that the steepest climb is perpendicular to that curve (the direction of the gradient), and that if if you were to set a ball on the ground and let it go, it would roll downhill perpendicular to that curve (in the direction of the negative of the gradient.)
The link I gave proves the theorem for a function of $3$ variables, but the same proof works in any number of dimensions of course.
answered Sep 6 at 6:07
saulspatz
11.6k21324
The link I gave has very good answers based on the directional derivative that prove the gradient points in the direction of steepest ascent, using directional derivatives. What has made this result intuitive for me is another result: the gradient is perpendicular to the level curve. Imagine standing on a hillside, and visualize the curve passing through all points at the same altitude as the point you're standing at. It just seems clear to me from experience, that the steepest climb is perpendicular to that curve (the direction of the gradient), and that if if you were to set a ball on the ground and let it go, it would roll downhill perpendicular to that curve (in the direction of the negative of the gradient.)
The link I gave proves the theorem for a function of $3$ variables, but the same proof works in any number of dimensions of course.
answered Sep 6 at 6:07
saulspatz
11.6k21324
answered Sep 6 at 6:07
saulspatz
11.6k21324
answered Sep 6 at 6:07
saulspatz
11.6k21324
answered Sep 6 at 6:07
saulspatz
11.6k21324
11.6k21324
add a comment |Â
add a comment |Â
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Possible duplicate of math.stackexchange.com/questions/223252/…
– saulspatz
Sep 6 at 5:44
You might also have a look at math.stackexchange.com/q/1912660/265466.
– amd
Sep 6 at 6:39