Proving the Divergence Formula for Plane Polars

In summary, the question asks to find the divergence of a vector field in plane polars. The given vector field is F(r,t) = Frer + Ftet where r and t are polar coordinates and er = (cos t, sin t, 0) and et = (- sin t, cos t, 0). The homework equations include converting polar coordinates to cartesian coordinates and the divergence formula in cartesian coordinates. The attempt at a solution involves partially differentiating the first and second components of the vector field, but the correct method uses the chain rule. The question has been successfully solved.
  • #1
Kate2010
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Homework Statement



I have to prove the divergence formula for plane polars. The question goes something like:

Find the divergence of the vector field F(r,t) = Frer + Ftet where r and t are polar coordinates and er = (cos t, sin t, 0) and et = (- sin t, cos t, 0)
(t is theta in the question but t was easier to type)


Homework Equations



x=rcost
y=rsint
Divergence formula in cartesian coordinates

The Attempt at a Solution



F(r,t) = (Frcost - Ftsint, Frsint + Ftcost, 0)

Could I partially differentiate the first bit with respect to r and the second bit with respect to t, just ignoring the 0 at the end? This does not seem right, I'm not sure if it is even possible.

Or I feel like the chain rule might come into it somewhere?

I really don't know where to start.
 
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  • #2
Solved it :)
 

FAQ: Proving the Divergence Formula for Plane Polars

What is divergence in plane polars?

Divergence in plane polars refers to the measure of how a vector field diverges or converges at a given point in a two-dimensional coordinate system. It is a scalar quantity that indicates the strength and direction of the vector field at that point.

How is divergence calculated in plane polars?

In plane polars, divergence is calculated using the polar coordinate system, which includes the radial distance and angle from the origin. The formula for divergence in plane polars is ∇·F = (1/r)(∂(rFr)/∂r + ∂Fθ/∂θ), where r is the radial distance and Fr and Fθ are the radial and angular components of the vector field F, respectively.

What does a positive or negative divergence value indicate?

A positive divergence value indicates that the vector field is spreading out or diverging at a given point, while a negative divergence value indicates that the vector field is converging or getting closer together at that point.

How is divergence related to the flow of a fluid?

Divergence is closely related to the flow of a fluid, as it represents the rate at which fluid is either expanding or contracting at a given point in space. In fluid dynamics, the divergence of a velocity field is used to determine the net flow of fluid out of or into a closed surface.

What are some real-world applications of divergence in plane polars?

Divergence in plane polars has several applications in fields such as fluid dynamics, electromagnetism, and image processing. It is used to analyze fluid flow and airfoil design, calculate electric and magnetic fields, and detect edges in images.

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