Understanding Vector Directional Derivatives

In summary, the formula for \nabla \times (\vec{A} \times \vec{B}) is equal to \vec{A}(\nabla \cdot \vec{B}) - \vec{B}(\nabla \cdot \vec{A}) + (\vec{B} \cdot \nabla)\vec{A} - (\vec{A} \cdot \nabla)\vec{B}. This expression involves operators such as \vec{\nabla} and dot products. The directional derivative is a scalar and dot products are commutative. The expression (\vec{A} \cdot \vec{\nabla}) is an operator that works on a vector
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I looked up the formula for [tex]\nabla \times (\vec{A} \times \vec{B})[/tex]:

[tex]\nabla \times (\vec{A} \times \vec{B}) = \vec{A}(\nabla \cdot \vec{B}) - \vec{B}(\nabla \cdot \vec{A}) + (\vec{B} \cdot \nabla)\vec{A} - (\vec{A} \cdot \nabla)\vec{B}[/tex]

What does a vector followed by a del mean? Mathworld says that in the context of a unit vector it's the directional derivative. It's unclear to me how this works because then you have a vector times a vector for the last two terms. Can someone please clarify how this works?
 
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  • #2
Edit: IGNORE!

[tex]{\vec A} \cdot \nabla = \nabla \cdot {\vec A}.[/tex]

The directional derivative is a scalar.
 
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Ah, ok. Dot products are commutative. Thanks.
 
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The expression [itex](\vec A \cdot \vec \nabla )[/itex] is an operator, like [itex]\vec \nabla[/itex]. It has to work on a vector. You can see what it does by treating [itex]\vec \nabla[/itex] as a genuine vector:

[tex]\vec A \cdot \vec \nabla = A_x \frac{\partial}{\partial x}+A_y \frac{\partial}{\partial y}+ A_z\frac{\partial}{\partial z}[/tex]

Ofcourse, you can always check it by writing out the components :P
 
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  • #5
oops, yeah, I'm insane! Though I should note that it can actually also work on a scalar.
 
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1. What is a vector directional derivative?

A vector directional derivative is a measure of how a scalar function changes in a specific direction in a vector field. It represents the rate of change of the function along a given direction in the field.

2. How is a vector directional derivative calculated?

The vector directional derivative is calculated using the dot product of the gradient of the function and a unit vector in the desired direction. This can be represented mathematically as Duf = ∇f · u, where Duf is the directional derivative, ∇f is the gradient of the function, and u is the unit vector in the desired direction.

3. What is the significance of the direction in a vector directional derivative?

The direction in a vector directional derivative represents the direction of the change in the function. It shows how the function is changing along a specific direction in the vector field.

4. How is the direction of maximum change determined in a vector directional derivative?

The direction of maximum change in a vector directional derivative is determined by the direction of the gradient of the function. The gradient points in the direction of steepest ascent, which is the direction of maximum change in the function.

5. What are some real-life applications of vector directional derivatives?

Vector directional derivatives are used in many fields, such as physics, engineering, and computer science. They can be used to analyze the flow of fluids, determine the direction of maximum heat transfer, and optimize the performance of algorithms, among other applications.

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