Vector Calculus: Operator Questions Answered

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I'm looking into vector calculus right now, and I have a few questions.

* is the dot operator

What is the difference between \nabla * F and F * \nabla ?

What is \nabla ^2 F, where F is a vector field?
 
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∇⋅F is ∂(F_x)/∂x + ∂(F_y)/∂y + ∂(F_z)/∂z which is a scalar function. F⋅∇ is (F_x)(∂/∂x) + (F_y)(∂/∂y) + (F_z)(∂/∂z) which is an operator.

∇²F for a vector field F is just the vector field with components ∇²(F_x), ∇²(F_y) and ∇²(F_z).
 
How did you get the symbols to work?
 
f*del can be thought of as the directional derivative in the direction of f
 
For original Zeta function, ζ(s)=1+1/2^s+1/3^s+1/4^s+... =1+e^(-slog2)+e^(-slog3)+e^(-slog4)+... , Re(s)>1 Riemann extended the Zeta function to the region where s≠1 using analytical extension. New Zeta function is in the form of contour integration, which appears simple but is actually more inconvenient to analyze than the original Zeta function. The original Zeta function already contains all the information about the distribution of prime numbers. So we only handle with original Zeta...

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