Help understanding the gradient

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In summary, the direction of maximum increase of f(x,y) is given by the gradient \nabla f(x,y), and the maximum value of D_{u}f(x,y) is ||grad f(x,y)||. The direction of maximum increase is -16i + 6j, and this is because moving in the negative x direction results in a positive change in the function value. This may seem counterintuitive based on the negative partial derivative, but the direction and gradient are separate concepts.
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Homework Statement


The evaluated partial derivative of f(x,y) with respect to x is -16 and 6 with respect to y at some point (x0,y0). What is the vector specifying the direction of maximum increase of f?


Homework Equations


The direction of maximum increase of f is given by [itex]\nabla[/itex]f(x,y). The maximum value of D[itex]_{u}[/itex]f(x,y) is ||grad f(x,y)||.


The Attempt at a Solution


I know the answer is -16i + 6j. But I just don't know why. I understand the geometric argument based on the dot product given in my textbook for why the gradient gives the direction of maximum increase. That's fine. But is there a more intuitive way to look at it in terms of partial derivatives so that the result can be easily extended to higher dimensions?

The main part that's confusing me is that the rate of change of f wrt x is negative (-16) and yet we want to move partly in the x direction. It seems like we would only want to move in the y direction since the rate of change of f wrt y is positive.

Any help would be much appreciated.
 
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  • #2
If the gradient is -16i+6j and you move in the NEGATIVE x direction then the function is increasing. A negative times a negative is positive.
 
  • #3
Thank you very much. I knew it was something easy like that.

I didn't see it because I kept going back to the limit definition of the derivative where it doesn't matter if you are interpreting the slope formula from the positive or negative direction.

It seems that direction doesn't matter until the calculation is done and then it becomes extremely important. Thus

dy/dx = 5 when x=2

is interpreted to mean that the rate of change of y as x varies in the positive direction from 2 is 5.
 
  • #4
Exactly. And if you are moving in the negative direction, it's -5. The gradient part and the direction part are separate.
 
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FAQ: Help understanding the gradient

1. What is the gradient?

The gradient is a mathematical concept that refers to the rate of change of a function at a specific point. It is represented by a vector or a set of partial derivatives, and it provides information about the direction and magnitude of the steepest ascent or descent of the function.

2. How is the gradient calculated?

The gradient is calculated by taking the partial derivatives of a multivariable function with respect to each of its variables. The resulting vector or set of partial derivatives represents the direction and magnitude of the steepest ascent or descent of the function at a specific point.

3. What is the significance of the gradient?

The gradient is significant because it provides important information about the behavior of a function at a specific point. It can help in determining the direction of maximum change or the direction in which a function increases or decreases the fastest. This information is useful in various fields, such as physics, engineering, and economics.

4. How can the gradient be visualized?

The gradient can be visualized as a set of vectors, with each vector pointing in the direction of the steepest ascent or descent of the function at a specific point. These vectors can be represented graphically using a slope field or a contour plot. Additionally, the gradient can also be represented as a set of elevation lines on a topographic map.

5. What is the relationship between the gradient and the slope?

The gradient and the slope are closely related concepts. The slope of a curve at a specific point is equal to the magnitude of the gradient vector at that point. Therefore, the gradient provides information about the slope of a curve in a specific direction, while the slope only represents the steepness of the curve in one direction.

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