The discussion revolves around finding the divergence of the vector field F(x,y,z) = (-x+y)i + (y+z)j + (-z+x)k. Participants are exploring the definition and calculation of divergence in the context of vector calculus.
The discussion is progressing with participants clarifying their understanding of divergence and the del operator. Some have recognized their misconceptions, and there is an exchange of insights regarding the correct approach to calculating divergence.
There is mention of confusion regarding the notation and definitions, particularly in distinguishing between the gradient and divergence. Participants are addressing these conceptual gaps without reaching a final resolution.
1MileCrash said:Homework Statement
F(x,y,z) = (-x+y)i + (y+z)j + (-z+x)k
Find divergence
Homework Equations
The Attempt at a Solution
The gradient is
-i + j + -k
Dotting that with F, I get
x - y + y + z + z - x
=
2z
My book lists the answer as -1. What the heck are they talking about? (they did not ask me to evaluate for any point)
Dick said:I think you are misunderstanding the definition of divergence. ∇.F doesn't mean grad(F).F. Look it up.
1MileCrash said:Is it correct to say that it's like taking grad F, then adding up the resulting components for a scalar?
1MileCrash said:I see my misunderstanding now. The del operator is not the gradient of anything in particular. It's just (d/dx)i + (d/dy)j + (d/dz)k. Dot product that with F leads to the correct definition.
This actually clears up a lot of the past notation. Since del is not a gradient of anything in particular, when we say \nabla f, since f is a scalar being multiplied by some vector, del, the result is a vector, which is the gradient of f.
Cool. :)