- #1
Jhenrique
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A scalar field can be the exact form of a vector field (potential form)? It's make sense?
Matterwave said:Nope, doesn't make sense. A divergence is not an exterior derivative. A scalar field is a 0-form. It can't be an exact form because an exact n-form must be the exterior derivative of of a n-1 form. There are no -1 forms, so a 0-form cannot be considered exact.
Matterwave said:You can't call it that. Those terms you used have very specific meanings. You call ##\vec{F}## the vector field and ##F## its divergence.
Jhenrique said:I'm speaking this way cause I have this ideia in my mind: http://en.wikipedia.org/wiki/Exact_form#Vector_field_analogies
F = ∇·F represents the relationship between a scalar field and a vector field, where the scalar field is given by F and the vector field is given by ∇·F. This equation is also known as the vector Laplacian operator.
A scalar field is a mathematical function that assigns a scalar value to every point in space. A vector field, on the other hand, is a mathematical function that assigns a vector to every point in space. Scalar fields have only magnitude, while vector fields have both magnitude and direction.
The gradient (∇) in F = ∇·F represents the rate of change of the scalar field F. It is a vector operator that points in the direction of the steepest increase of the scalar field at a given point.
F = ∇·F is used in various physical and engineering applications, such as fluid dynamics, electromagnetism, and heat transfer. It helps to understand the behavior of scalar and vector fields, and their interactions with each other.
One example of F = ∇·F in real life is the flow of water in a river. The scalar field represents the water level at different points along the river, while the vector field represents the direction and speed of the water flow. By understanding the relationship between these two fields, we can predict the movement of water and make informed decisions about flood control and navigation.