The vector[s] pointing in the direction of greatest increase are those with the largest relative "vertical" components. Plane geometry in the tangent plane shows that these must to be perpendicular. If gradient is not perpendicular to the level curve, it will have some component along the level curve. This means that function's value will increase if you move in that direction, but on a level curve, function's value can not increase or decrease so gradient can not have component along level curve and this is possible only when gradient is perpendicular to level curve.
A very useful way to think about the gradient or more generally, the first derivative of any function on some Euclidean space is as "the thing that gives you the best linear approximation to the function at a given point.
That generalizes, by the way, to vector-valued functions: then f' p is a linear transformation, not simply a vector. The point is that if you look at your function very closely near the point p, then it looks more and more like that linear function, and the approximation just gets better as you look closer. In particular, that linear approximation completely captures both the direction of the level curve of f through p and the direction of fastest growth.
Now you should visualize a linear function of two variables, whose graph is simply a slanted plane, and it should be obvious that the level curves which are horizontal lines embedded in the plane are perpendicular to the slant of the plane, if you're visualizing it correctly.
One subtlety that my explanation doesn't completely cover is what happens when the gradient is zero: that would give you the linear approximation.
Of course, the function f may not literally be constant near p, so you might still want to know what the level set looks like and where the fastest growth is.
To get that information, you will have to use the higher derivatives. All the gradient tells you is that:. Start in the 1-variable case.
The derivative of a function at a point is a number that can be considered a vector that either points left or right. It is by default perpendicular to the level curve which is a point. So the perpendicular to the tangent of that graph f' c ,-1 is up to a constant the gradient of g. Finally, consider the case of a linear function. The proof I usually see: Choose an arbitrary unit length tangent vector on the level set, and write it with coordinates.
If you take the inner product of this vector with the gradient, the sum you get is the definition of the directional derivative along that vector, and therefore zero. I think there is a more geometric way to think of it by taking a linear approximation to your function near a noncritical point, and restricting the linear function to a sphere. After a suitable rotation, the extrema lie at poles, and the level set lies at the equator.
Sign up to join this community. The best answers are voted up and rise to the top. Why is the gradient normal? Ask Question. Asked 12 years ago. Active 8 months ago. Viewed 71k times. I have always struggled to find the correct internal model that would encapsulate these ideas. Improve this question. Rodrigo de Azevedo 2, 3 3 gold badges 13 13 silver badges 28 28 bronze badges. Kim Greene Kim Greene 3, 10 10 gold badges 39 39 silver badges 39 39 bronze badges. The heart of what's going on here is the isomorphism between a vector space and its dual induced from an inner product.
That leads immediately to the interpretation. Add a comment. Active Oldest Votes. Improve this answer. G 3, 5 5 gold badges 25 25 silver badges 53 53 bronze badges.
And the question is when is this maximized? What unit vector maximizes this? And if you start to imagine maybe swinging that unit vector around, so if, instead of that guy, you were to use one that pointed a little bit more closely in the direction, then it's projection would be a little bit longer.
Maybe that projection would be like 0. If you take the unit vector that points directly in the same direction as that full vector, then the length of its projection is just the length of the vector itself. It would be one, because projecting it doesn't change what it is at all.
So it shouldn't be too hard to convince yourself, and if you have shaky intuitions on the dot product, I'd suggest finding the videos we have on Khan Academy for those. Sal does a great job giving that deep intuition.
It should kind of make sense why the unit vector that points in the same direction as your gradient is gonna be what maximizes it, so the answer here, the answer to what vector maximizes this is gonna be, well, it's the gradient itself, right? It is that gradient vector evaluated at the point we care about, except you'd normalize it, right? Because we're only considering unit vectors, so to do that, you just divide it by whatever it's magnitude is.
If its magnitude was already one, it stays one. If its magnitude was two, you're dividing it down by a half. So this is your answer. This is the direction of steepest ascent. So, I think one thing to notice here is the most fundamental fact is that the gradient is this tool for computing directional derivatives. You can think of that vector as something that you really want to dot against, and that's actually a pretty powerful thought, is that the gradient, it's not just a vector, it's a vector that loves to be dotted together with other things.
That's the fundamental. And as a consequence of that, the direction of steepest ascent is that vector itself because anything, if you're saying what maximizes the dot product with that thing, it's, well, the vector that points in the same direction as that thing.
And this can also give us an interpretation for the length of the gradient. We know the direction is the direction of steepest ascent, but what is the length mean? So, let's give this guy a name. Let's give this normalized version of it a name. I'm just gonna call it W. So W will be the unit vector that points in the direction of the gradient.
If you take the directional derivative in the direction of W of f, what that means is the gradient of f dotted with that W. And if you kind of spell out what W means here, that means you're taking the gradient of the vector dotted with itself, but because it's W and not the gradient, we're normalizing.
We're dividing that, not by magnitude of f, that doesn't really make sense, but by the value of the gradient, and all of these, I'm just writing gradient of f, but maybe you should be thinking about gradient of f evaluated at a,b, but I'm just being kind of lazy, and just writing gradient of f. And the top, when you take the dot product with itself, what that means is the square of its magnitude.
But the whole thing is divided by the magnitude. So you can kind of cancel that out. You could say this doesn't need to be there, that exponent doesn't need to be there, and basically, the directional derivative in the direction of the gradient itself has a value equal to the magnitude of the gradient.
So this tells you when you're moving in that direction, in the direction of the gradient, the rate at which the function changes is given by the magnitude of the gradient. I've been thinking on this for awhile now well, I first learned of this fact about 15 years ago, so I'd say awhile , and I now had a bit of a thought.
So what does the tangent plane look like? These two vectors certainly span a two dimensional space, so they take care of our whole tangent plane, and our gradient really is orthogonal to it.
I don't know if this is the answer you need, but this is the explanation I'm satisfied with. Happy to answer any follow up questions, or receive any criticism. Steven Gubkin's excellent answer contains a proper explanation. If then you have instead three variables, and thus level surfaces, the "visual representation" above does not change in substance.
Simply put: The gradient at a point p a,b is the greatest rate of change of z a,b. The level curve has constant value z. Therefor for these two "lines" to satisfy there definition, they must be perpendicular to one another. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Why gradient vector is perpendicular to the plane Ask Question.
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