Understanding Scalar Fields and the Laplace Equation: How Do They Relate?

[danong]

I've recently read about Null Identities of vector analysis.
I'm having a problem in understanding what is it by "taking the curl of the grad of any scalar field is equal to zero."

What is by definition of scalar field then? How would it looks like? Is position vector a scalar field? If No, then What's the difference between them?

For say if i have a position field P, then by taking partial differentiation i achieve V (grad of P), which by means if i take the curl of V, does that means it is always irrotational no matter what?

Thanks in advance.

 

[nicksauce]

A scalar field is a function that maps R^n to R.

Example:
$$\Phi = x^2 + y^2 + sin(z)$$

Contrast that with a vector field which might look like:
$$F = x^2\vec{i} + y^2\vec{j} + sin(z)\vec{k}$$

 

[elibj123]

scalar field is a function of a vector that returns a scalar.
The gradient of a scalar field is a vector field that points towards the maximal slope of the function at every point.

curl(grad(f))=0 means that every vector field which can be derived of a scalar function (also called: potential, a name borrowed from physics) is a conservative field.

 

[danong]

alright thanks for the explanation =)

 

[danong]

Sorry but i still have a question regarding Laplace Equation,
say if a potential function P represents the inverse square propotional field,
then how am i going to visualize taking twice partial derivative of P (xi, yj, zk) is equal to zero?

Because since grad of P is pointing inward (which looks to me is a sink at the centre of the gravitation point (a0,b0,c0)),

so how am i going to say that divergence of grad(P) is equal to zero? (since the grad(P) the vector is pointing inward),

I mean i have seen some proofs of it which leads to the final conclusion of Laplace Equation,
but how am i going to visualize it in a way that it makes sense that grad(P) is pointing no-where? (since divergence of grad(P) should be zero, which means neither sink or source, but isn't gravitation a sink? ).