Math Problem Statement
Solution
To tackle the problem in the image, here’s the breakdown and explanation:
Part (a): Prove that the force field is source-free in Cartesian coordinates
The force field is given as: or equivalently: where .
Step 1: Expressing the divergence
The divergence in Cartesian coordinates is: where the components of are:
Step 2: Compute
Using the product rule and chain rule: Let , so . Using this:
A similar computation applies to and .
Step 3: Summing the terms
Adding these components, all terms proportional to cancel out, leaving: which shows that the field is source-free ( has no divergence).
Part (b): Express and calculate divergence in spherical coordinates
In spherical coordinates, the force field is:
Step 1: Divergence in spherical coordinates
The divergence in spherical coordinates for a radial field is given by: where .
Step 2: Compute divergence
Simplify: and: Thus: showing that the field remains source-free in spherical coordinates.
Summary
- The force field is source-free in Cartesian coordinates ().
- Expressing the force in spherical coordinates and calculating divergence also gives , consistent with the source-free nature of the gravitational field.
Would you like additional detailed steps or visualizations for these calculations?
Related Questions:
- How does the divergence theorem verify that the gravitational field is source-free?
- Can we generalize this proof to other inverse-square law forces (e.g., electric field)?
- What role does the delta function play in defining divergence at the origin?
- How is the expression for different in cylindrical coordinates?
- What physical significance does the divergence of zero have in gravitational systems?
Tip: Always cross-check calculations for divergence in Cartesian vs.
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Math Problem Analysis
Mathematical Concepts
Vector Calculus
Divergence of a Vector Field
Spherical Coordinates
Cartesian Coordinates
Formulas
F = -(GMm / (x^2 + y^2 + z^2)^(3/2))(xi + yj + zk)
F = -(GMm / r^2)e_r
Divergence in Cartesian: ∇·F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z
Divergence in Spherical: ∇·F = (1/r^2)(∂/∂r)(r^2 Fr)
Theorems
Divergence Theorem
Source-Free Field in Gravitational Physics
Suitable Grade Level
Undergraduate Physics or Mathematics
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