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 [itex]\nabla f[/itex], since f is a scalar being multiplied by some vector, del, the result is a vector, which is the gradient of f.
Cool. :)
The divergence of a vector field is a measure of the flow or flux of a vector field at a given point in space. It represents the amount of "outwardness" or "inwardness" of the vector field at that point.
Divergence is calculated by taking the dot product of the vector field with the del operator (∇). This is represented by the formula ∇ · F, where F is the vector field.
A positive divergence value indicates that the vector field is spreading out from a given point, while a negative divergence value indicates that the vector field is converging towards a given point.
Divergence of a vector field is used in many fields of science and engineering, such as fluid dynamics, electromagnetism, and meteorology. It can be used to analyze the flow of fluids, the behavior of electric and magnetic fields, and the movement of air masses in weather patterns.
Divergence and curl are both measures of vector fields, but they represent different properties. Divergence measures the flow or flux of a vector field, while curl measures the rotation or circularity of a vector field. In some cases, the divergence and curl of a vector field can be used together to fully describe the behavior of the field.