Which gradient is the steepest

Substitution 2. Powers of sine and cosine 3. Trigonometric Substitutions 4. Integration by Parts 5. Rational Functions 6. Numerical Integration 7. Additional exercises 9 Applications of Integration 1. Area between curves 2. Distance, Velocity, Acceleration 3. Volume 4. Average value of a function 5. Work 6. Center of Mass 7. Kinetic energy; improper integrals 8. Probability 9. Arc Length Polar Coordinates 2. Slopes in polar coordinates 3. Areas in polar coordinates 4. Parametric Equations 5.

Calculus with Parametric Equations 11 Sequences and Series 1. Sequences 2. Series 3. The Integral Test 4. Alternating Series 5. Comparison Tests 6. Absolute Convergence 7. The Ratio and Root Tests 8. Power Series 9. Calculus with Power Series Taylor Series Taylor's Theorem Additional exercises 12 Three Dimensions 1.

Lemma 1. The formula we derived looks very much like the Pythagorean theorem, and indeed, the latter is the special case of the former. To prove it, we make use of the fact that by definition, the angle between two vectors is the same as the angle between the sides of a triangle that corresponds to these vectors. This is illustrated in the following figure:. If you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Donate Login Sign up Search for courses, skills, and videos. Math Multivariable calculus Derivatives of multivariable functions Gradient and directional derivatives. Practice: Finding gradients. Practice: Visual gradient. Gradient and contour maps.

Directional derivative, formal definition. Practice: Finding directional derivatives. Directional derivatives and slope. Why the gradient is the direction of steepest ascent. Next lesson. Current timeTotal duration Google Classroom Facebook Twitter. Video transcript - [Voiceover] So far, when I've talked about the gradient of a function, and let's think about this as a multi-variable function with just two inputs.

Those are the easiest to think about. So maybe it's something like x squared plus y squared, a very friendly function. When I've talked about the gradient, I've left open a mystery.

We have the way of computing it, and the way that you think about computing it is you just take this vector, and you just throw the partial derivatives in there.

Partial with respect to x, and the partial with respect to y, and if it was a higher dimensional input, then the output would have as many variables as you need. If it was f of x,y,z, you'd have partial x, partial y, partial z. And this is the way to compute it.

But then I gave you a graphical intuition. I said that it points in the direction of steepest ascent, and maybe the way you think about that is you have your input space, which in this case is the x,y plane, and you think of it as somehow mapping over to the number line, to your output space, and if you have a given point somewhere, the question is, of all the possible directions that you can move away from this point, all those different directions you could go, which one of them-- this point will land somewhere on the function, and as you move in the various directions maybe one of them nudges your output a little bit, one of them nudges it a lot, one of it slides it negative, one of them slides it negative a lot.

I hope that helps. It took me a while to understand this intuitively and the image of a diagonal within a rectangle finally switched the lamp on.

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 is gradient the direction of steepest ascent? Ask Question. Asked 9 years ago. Active 3 months ago. Viewed 68k times.

Why do I get maximal value again if I move along with the direction of gradient? Rodrigo de Azevedo Jing Jing 1, 3 3 gold badges 16 16 silver badges 25 25 bronze badges. Add a comment. Active Oldest Votes. How do you know there is not other vector that moving in its direction might lead to a steeper change?

How does that answer the question? Show 3 more comments. It is simply a rate of change. I like your explanation overall regardless.