Vector Calculus: Operator Questions Answered

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In summary, vector calculus includes the dot operator, which is represented by ∇⋅F and F⋅∇. The difference between these two is that ∇⋅F is a scalar function, while F⋅∇ is an operator. ∇²F, for a vector field F, is simply the vector field with components ∇²(F_x), ∇²(F_y), and ∇²(F_z). In other words, it is the second derivative of each component of the vector field. Additionally, f*del can be understood as a directional derivative in the direction of f, while del*f is the gradient of f.
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I'm looking into vector calculus right now, and I have a few questions.

* is the dot operator

What is the difference between \nabla * F and F * \nabla ?

What is \nabla ^2 F, where F is a vector field?
 
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  • #2
∇⋅F is ∂(F_x)/∂x + ∂(F_y)/∂y + ∂(F_z)/∂z which is a scalar function. F⋅∇ is (F_x)(∂/∂x) + (F_y)(∂/∂y) + (F_z)(∂/∂z) which is an operator.

∇²F for a vector field F is just the vector field with components ∇²(F_x), ∇²(F_y) and ∇²(F_z).
 
  • #3
How did you get the symbols to work?
 
  • #4
f*del can be thought of as the directional derivative in the direction of f
 

1. What is vector calculus?

Vector calculus is a branch of mathematics that deals with the differentiation and integration of vector fields, also known as functions that have both magnitude and direction.

2. What are the basic concepts of vector calculus?

The basic concepts of vector calculus include vector fields, line integrals, surface integrals, and the fundamental theorem of calculus for line and surface integrals.

3. How is vector calculus used in real-world applications?

Vector calculus is used in many fields such as physics, engineering, and computer graphics. It can be used to describe and analyze physical phenomena like fluid flow, electromagnetism, and motion in three-dimensional space.

4. What are some common operators used in vector calculus?

Some common operators used in vector calculus include gradient, divergence, curl, and Laplacian. These operators are used to manipulate and analyze vector fields.

5. Is vector calculus difficult to learn?

Vector calculus can be challenging, but with practice and a solid understanding of basic mathematical concepts, it can be mastered. It is important to have a strong foundation in algebra, trigonometry, and calculus before diving into vector calculus.

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