Directional derivative and gradient definition confusion

[thegreengineer]

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.

 

[Orodruin]

I would suggest starting in a standard textbook or on Wikipedia:
http://en.wikipedia.org/wiki/Gradient
http://en.wikipedia.org/wiki/Directional_derivative
There is a significant amount of information on both the gradient and the directional derivative in these links. When you have read what they have to say, we will be happy to help you iron out any question marks that arise in relation to this.

 

[thegreengineer]

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.

 

[Orodruin]

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.

 

[AMenendez]

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.

 

[thegreengineer]

Thanks

 

[jtbell]

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).