Jan 12 2011

Gradient of a Scalar Potential

In Thomas’ Calculus we find the definition: “If F is a vector field defined on D and F = del f for some scalar function f on D, then f is called a potential function for F.” ([1], pg 921)

Kaplan states that the “gravitational field is the gradient of the scalar f = kMm/r” ([2], Prob. 4, pg 180), which for General Relativity is indeed the case.  With gamma ray energy exchange however, the scalar is f = k1M/r, in form much like the scalar potential k2q/r with electrodynamics.  This tells us that both gravitational and Coulomb forces are of independent action, and the corresponding fields are gradients of scalar potentials, which can be considered as more evidence for unification.

The vector fields we are looking at have units of N/kg for gravity, and of course E = N/C for an electric field.  A gravitational field calculation is most applicable when the object in the field is small compared to, or far away from, the source of the field, such as Jupiter compared to the sun.

Aside from the King James Version of the Bible, Wilfred Kaplan’s Advanced Calculus is still my favorite book.  By the way, we used Thomas’ Calculus book at the University of Wisconsin in 1976 and 1977.

 

[1] Thomas, George B., as revised by Weir, Maurice D. and Hass, Joel, Thomas’ Calculus, Twelfth Edition, Addison-Wesley of Pearson Education, Inc., 2010, 2005, 2001

[2] Kaplan, Wilfred, Advanced Calculus, Fifth Edition, Addison-Wesley of Pearson Education, Inc., 2003

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