Directional derivative and gradient definition confusion

In summary: If you choose some other direction, you'll get some intermediate value.In summary, the conversation discusses the concepts of multivariable calculus such as multivariable function, partial derivative, directional derivative, and gradient. The directional derivative is defined as the change in a function for small changes in the argument in a specific direction, while the gradient describes the direction of steepest slope of a function. These concepts are useful in describing physical phenomena and can be found in standard textbooks or on Wikipedia.
  • #1
thegreengineer
54
3
Recently I started with multivariable calculus; where I have seen concepts like multivariable function, partial derivative, and so on. A week ago we saw the following concept: directional derivative. Ok, I know the math behind this as well as the way to compute the directional derivative through the dot product of the gradient and the unit vector:

Duf(x,y)=∇f(x,y)⋅u​
Where:
  • Duf(x,y) represents the directional derivative
  • ∇f(x,y) represents the gradient
  • u represents the unit vector
Ok, I know the equation, but my questions are the following: what's exactly a directional derivative? what's a gradient? for what are they used?

Thanks.
 
Mathematics news on Phys.org
  • #2
  • #3
Dude, these links only say what they are in equations! I mean, for example, we can define the integral as the area below the curve of a function f, the derivative is a tangent line in a point in a graph, I'm asking how do we physically define a directional derivative and a gradient.
 
  • #4
The gradient and directional derivative are mathematical objects. That they are very useful in describing physical phenomena is a different matter.

The directional derivative is defined just as you would expect, how the function changes for small changes of the argument in a certain direction. The gradient describes the direction in which a function changes the most and how fast it does so.
 
  • #5
The idea behind the directional derivative (as opposed to a partial derivative) is that we ideally want to know how the function ##f(x, y)## is changing with respect to both ##x## and ##y##. The problem is that ##x## and ##y## may be changing at different rates, affecting the "direction", if you will.

The gradient is a vector comprised of a function's partial derivatives and is always orthogonal to a level surface (##f(x, y, z) = k##) at a point.
 
  • #6
Thanks
 
  • #7
MarcusAu314 said:
what's exactly a directional derivative? what's a gradient?

Using a concrete example, let f(x,y) specify the height (elevation) of points (x,y) on the ground, in a hilly landscape.

The gradient is a vector, defined at each point (x,y). The direction of the vector tells you the direction of steepest uphill slope at that point. The magnitude of the vector tells you the value of that steepest uphill slope.

The directional derivative tells you the slope at point (x,y) along a direction which you specify. If that direction happens to be the direction of the gradient vector, the directional derivative is just the magnitude of the gradient vector, i.e. the steepest slope at that point. If the direction is perpendicular to the gradient vector, the directional derivative should be zero (you're going "sideways" around the hill at constant elevation).
 

1. What is the difference between directional derivative and gradient?

The directional derivative is a measure of the rate of change of a function in a specific direction, while the gradient is a vector that points in the direction of the steepest increase of a function.

2. How do you calculate the directional derivative and gradient of a function?

The directional derivative is calculated by taking the dot product of the gradient vector and a unit vector in the desired direction. The gradient is calculated by taking the partial derivatives of the function with respect to each variable and organizing them into a vector.

3. Can the directional derivative and gradient be negative?

Yes, both the directional derivative and gradient can be negative. The sign indicates the direction of decrease, while the magnitude represents the rate of decrease.

4. What is the significance of the directional derivative and gradient in real-world applications?

The directional derivative and gradient are important concepts in fields such as physics, engineering, and economics. They help us understand the rate of change of a function and the direction of steepest increase, which can be used to optimize processes and make predictions.

5. Can you explain the concept of gradient descent using the directional derivative and gradient?

Gradient descent is an optimization algorithm that uses the gradient to find the minimum or maximum of a function. It works by taking small steps in the direction of the negative gradient, which leads to the steepest decrease of the function. This process continues until a minimum or maximum is reached.

Similar threads

Replies
4
Views
772
  • Calculus and Beyond Homework Help
Replies
8
Views
302
  • Science and Math Textbooks
Replies
10
Views
2K
Replies
9
Views
646
  • Calculus and Beyond Homework Help
Replies
4
Views
502
  • Differential Geometry
Replies
9
Views
332
  • General Math
Replies
1
Views
593
Replies
18
Views
1K
  • Advanced Physics Homework Help
Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
Back
Top