A quick check-up on directional derivatives

Thread starter Char. Limit
Start date Dec 12, 2010
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Derivatives

In summary, a directional derivative is a measure of how a function changes in a given direction at a specific point. It is calculated using the dot product of the gradient and the unit vector in the desired direction. Directional derivatives are significant in multivariate calculus and optimization problems, and can be negative if the function is decreasing in the given direction. The direction of steepest increase is related to the directional derivative as it corresponds to the direction of the gradient vector at that point.

Dec 12, 2010

Char. Limit

Gold Member

Just a quick question...

To calculate a directional derivative of f(x,y) at the point [tex]\vec{u}[/tex] in the direction [tex]\hat{v}[/tex], can I just use the formula...

[tex]\nabla f(\vec{u}) . \hat{v}[/tex]?

It would be so easy.

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Dec 12, 2010

Astronuc

Staff Emeritus

Science Advisor

2023 Award

Hint - http://mathworld.wolfram.com/DirectionalDerivative.html

Dec 12, 2010

Char. Limit

Gold Member

Thanks!

FAQ: A quick check-up on directional derivatives

What is a directional derivative?

A directional derivative is a measure of how a function changes in the direction of a given vector at a specific point. It represents the rate of change of the function in that direction.

How is a directional derivative calculated?

A directional derivative is calculated using the dot product of the gradient of the function and the unit vector in the desired direction. This can also be written as the product of the magnitude of the gradient and the cosine of the angle between the gradient and the direction vector.

What is the significance of directional derivatives?

Directional derivatives are important in multivariate calculus as they allow us to understand how a function changes in a specific direction at a given point. They also play a crucial role in optimization problems and the study of surfaces and curves in higher dimensions.

Can a directional derivative be negative?

Yes, a directional derivative can be negative. This indicates that the function is decreasing in the direction of the given vector at the specified point.

How is the direction of steepest increase related to directional derivatives?

The direction of steepest increase is the direction in which the directional derivative is maximum. This corresponds to the direction of the gradient vector at that point.