Dot product of vector and del.

In summary, the conversation discusses the confusion surrounding the expression (u $ ∇) and whether it is the same as (∇ $ u). It also clarifies that the symbols used represent the dot product and the vector differentiation operator. The usual convention is that ∇ acts to the right, making (u $ ∇) and (∇ $ u) not equal. The analogy of this question is also provided.
  • #1
pyroknife
613
3
I'm not sure which section is best to post this question in.

I was wondering if the expression (u $ ∇) is the same as (∇ $ u).
Here $ represents the dot product (I couldn't find this symbol.
∇=del, the vector differentiation operator
and u is the velocity vector or any other vector
 
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  • #2
The usual convention is that ∇ acts to the right so (u $ ∇) and (∇ $ u) are not equal.

This is analogous to asking if uD is equal to D u where D is the differentiation operator.
 
  • #3
QUOTE=pyroknife;4654871]I'm not sure which section is best to post this question in.

I was wondering if the expression (u $ ∇) is the same as (∇ $ u).
Here $ represents the dot product (I couldn't find this symbol.
∇=del, the vector differentiation operator
and u is the velocity vector or any other vector[/QUOTE]
Before anyone can answer that question, you will have to tell us what you mean by "(u $ ∇). The reason I say that is that things like [itex]\nabla\cdot u[/itex] and [itex]\nabla\times u[/itex] are mnemonics for [itex]\partial u_x/\partial x+ \partial u_y/\partial y+ \partial u_z/\partial [/itex] and [itex](\partial u_z/\partial y- \partial u_y/\partial z)\vec{i}+ (\partial u_x/\partial z- \partial u_z/\partial x)\vec{j}+ (\partial u_y/\partial x- \partial u_x/\partial y)\vec{k}[/itex]. In particular "[itex]\nabla[/itex]" is NOT a real vector and you cannot combine it with vector functions without saying HOW that is to be done.
 

1. What is the dot product of a vector and del?

The dot product of a vector and del is a mathematical operation that results in a scalar value. It is calculated by multiplying the components of the vector with the corresponding partial derivatives of the del operator and adding them together. This operation is also known as the scalar product or inner product.

2. How is the dot product of a vector and del used in physics?

In physics, the dot product of a vector and del is used to calculate important quantities such as work, energy, and flux. For example, in electromagnetics, the dot product is used to find the electric field produced by a charge distribution. In fluid dynamics, it is used to calculate the rate of change of a physical quantity over a given area.

3. What is the geometric interpretation of the dot product of a vector and del?

The dot product of a vector and del can be interpreted as the projection of one vector onto the other. In other words, it represents the component of one vector in the direction of the other. This geometric interpretation is useful in understanding the physical significance of the dot product in various applications.

4. How does the dot product of a vector and del relate to the gradient?

The gradient is a mathematical operator that represents the rate of change of a scalar field. The dot product of a vector and del is closely related to the gradient, as it can be used to calculate the directional derivative of a scalar field in a specific direction. This relationship is important in many areas of physics and engineering.

5. What are some real-life applications of the dot product of a vector and del?

The dot product of a vector and del has many practical applications in fields such as engineering, physics, and computer science. Some examples include calculating the flow of heat or fluid in a system, finding the electric field in electromagnetics, and optimizing algorithms in machine learning. It is also used in 3D graphics and computer animation to calculate the lighting and shading of objects.

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