Understanding Divergence: Can You Find the Divergence of a Scalar Function?

In summary, the lecturer has provided a scalar function g(x,y,z) and vector field F, and has asked to find the gradient and divergence of g. However, it seems that the request to find the divergence of g is a typo, as the divergence is only applicable to vector fields and not scalar functions. This was confirmed by another individual, Daniel, who explained that the divergence is a differential operator that reduces the rank of a tensor by one, but since the scalar function already has a rank of 0, it does not make sense to apply the divergence.
  • #1
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I'm doing an assignment where the lecturer has said scalar function g(x,y,z) = x^3 + y + z^2

and vector field F = (2xz,sin y,e^y)

and asked find


a) grad g


which is fairly easy, but then

b) div g

and my understanding was that you can only find the divergence of a vector field not a scalar function.

Am I right an there's been a typo and he meant div F, or can you actually find div g, Because there's no actually mention of F in any of the questions, which is odd.
 
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  • #2
You're entirely right. It's a typo.
 
  • #3
The scalars don't have a divergence.The divergence is a differntial operator which,applied on a tensor of rank "n",reduces the rank by a unit,namely to "n-1".Since the scalar has already rank "0",you see that it makes no sense to apply the divergence.

Daniel.
 

What is Vector Calculus?

Vector Calculus is a branch of mathematics that deals with the study of vector fields, which are functions that assign a vector to each point in a given space. It involves the application of calculus concepts, such as differentiation and integration, to vectors.

What are some common problems encountered in Vector Calculus?

Some common problems encountered in Vector Calculus include finding the gradient, divergence, and curl of a vector field, solving line and surface integrals, and applying the theorems of Green, Stokes, and Gauss.

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Vector Calculus is used in a variety of real-life applications, such as physics, engineering, economics, and computer graphics. It is particularly useful in modeling and analyzing physical systems, such as fluid flow, electromagnetism, and heat transfer.

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Some useful tools for solving problems in Vector Calculus include the gradient, divergence, and curl operators, as well as the theorems of Green, Stokes, and Gauss. Additionally, software programs like Mathematica and MATLAB can be helpful in visualizing and solving complex vector calculus problems.

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