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Polar Derivative, Coordinates & Function

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Contents:

  1. Polar Coordinates Definition
  2. What is a Polar Function?
  3. Polar Derivatives

Polar Coordinates Definition

Polar coordinates are very similar to the “usual” rectangular coordinates: both systems are two dimensional, they locate a point in space, and both use two points: the rectangular system uses (x, y) and the polar coordinate system uses (r, θ).

Plotting Polar Coordinates

To plot polar coordinates, you need two pieces of information, r and θ:

  • θ tells you the ray’s angle from the polar axis (the positive part of the x-axis).
  • “r” tells you how to move on the ray. If r > 0, move on the ray. If r < 0, move on the opposite ray.

The easiest way to understand how to plot polar coordinates is to plot one first on the familiar x-y axis. Let’s say you wanted to plot point P, located at (4, 15°). Here, r = 3, so that’s going to be equal to x = 3. The angle of the ray from the x-axis is 15 degrees, so the goal is to draw the 15 degree ray, then locate the point:

polar coordinates

The green ray is at 15 degrees from the x axis. The point is at x = 4, along that ray.



What about (-4, 15°)? To get a negative “x” value, just move in the opposite direction, just like you’d do with regular coordinates.
negative values polar coordinates

The whole grid looks like this:
polar coordinate system

In the above example, the negative number in (-4, 15°) meant that you travel in the opposite direction from the 15 degree ray. You could also write the exact same point as (-4, 195°).

Why Use Polar Coordinates?

Polar coordinates make it easier to understand some natural phenomena with circular motion from a central point, like the motion of planets around the sun or atoms around a nucleus.

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Polar Function Definition

A polar function relates a radius vector (a distance, r) and an angle vector (θ). These functions are based in the polar coordinate system.

doppler radar

Doppler radar uses polar (circular) coordinates.

Many natural phenomena can be represented by a polar function. For example:

  • Anywhere objects move in circles (for example: movement of electrons),
  • A plan position indicator (a type of radar display) used in air traffic control, ship navigation and meteorology,
  • Radiance functions for material brightness can be represented by a polar function on the unit sphere (Robles-Kelly & Hancock, 2004).

Types of Polar Function

A polar function is defined by the polar equation r = f(θ). Many polar functions have been classified in detail, including:

1. Lemniscate Polar Function

A polar function of the form r2 = a2 sin (2θ) and r2 = a2 cos (2θ) are lemniscates (from the Latin lēmniscātus meaning decorated with ribbons).

lemniscate

Graph of the lemniscate r2=22sin(2θ).


Click here for an interactive version of this graph on Desmos.com.

2. Limaçon & Cardioid

Limaçons (from the Latin limax meaning snail) are formed by the following equations:

  • r = a + b sin θ,
  • r = a – b sin θ,
  • r = a + b cos θ,
  • r = a – b cos θ.

Cosine and sine are shifted 45 degrees from each other. For example:

Limaçon

Limaçon polar functions r = 2 + 3 cosθ (red) & r = 2 + 3 cosθ


Click here for an interactive version of the graph.

A cardioid (a heart shaped curve) is a special case of the Limaçon, graphed when a = b.

cardiod polar function

The cardiod r = 2 – 2 sin(θ).

3. Rose Polar Function

A polar function with the form r = a sin nθ or r = a cos nθ graph roses. The two functions look almost identical, except they are shifted:

rose polar equation

Two rose functions: r = 2 cosθ (red) and r = 2 sinθ (blue)


For an interactive graph (where you can change the values for r and θ) click here to go to Desmos.

4. The Archimedean Spiral

The Archimedean spiral is defined by r = aθ.
polar function

For an interactive graph, go to: Desmos.com.

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Polar Derivatives

The polar derivative generalizes the usual derivative to polar coordinates. In other words, the derivative rules you used in elementary calculus only work in the Cartesian plane. In order to find the derivative of a polar function, you have to use a different formula.

As polar coordinates are based on angles, it should be no surprise that derivatives involve a little trigonometry. A polar coordinate can be expressed in terms of:

  1. The distance from the origin (r) and
  2. An angle (θ).

The first derivative of a polar curve uses these coordinates in the formula:

Polar Derivative: Example Problem

There are a couple of ways to find a polar derivative. The first is to use the above formula. However, instead of memorizing yet another formula, you could convert your coordinates and use the product rule instead; The following example shows how this method works to get the same result.

Example Question: Find the polar derivative of r = 2 sin(θ) at π/2.

Step 1: Convert the polar coordinates (r, θ) to Cartesian coordinates. The conversion formulas are:

  • x = r cos(θ)
  • y = r sin(θ)

We’re given r = 2 sin(θ) in the question, so:

  • x = (2 sinθ) cos(θ)
  • y = (2 sinθ) sin(θ)

Step 2: Find the derivative of y from Step 1. The function y = (2 sinθ) sin(θ) is two functions multiplied together, so for this example, use the product rule: (f * g)′ = f′ * g + f * g′.

Inserting the value for “y” from Step 1 into the product rule formula, we get:
(2 sin θ)(cos θ) + (sin θ)(2 cosθ)

Step 3: Find the derivative of x from Step 1. The function x = (2 sinθ) cos(θ) can be differentiated with the product rule as well, so:
(2 sin θ)(-sin θ) + (cos θ)(2 cosθ)

Step 4: Divide Step 2 (dy) by Step 3 (dx), to get:
step 4 polar derivative

Step 5: Insert your value for θ, which is given in the question as π/2:
polar coordinate derivative step 5

Step 6: Simplify, using a calculator to find values. For example, cosθ = 0.
how to find a polar derivative

Solution: The derivative is 0.

That’s it!

Polar Derivative and Complex Numbers

The polar derivative can be defined for complex numbers as:


Dα p(z) = np(z) + (α – z) p′(z);

Where α = a complex number and p ∈ Pn (Li, 2011).

There are other slightly different notations. For example, you could write the polar derivative with respect to zeta(ζ):
fζ(z): = nf(z) + (ζ – z) f′(z),
where ζ is a complex number.
If degree f(z) = n, then fζ(z) is a polynomial function with degree n – 1. If ζ = ∞, then f is equal to the ordinary derivative (Barsegian et al., 2006).

References

Barsegian, G. et al., (2006). Value Distribution Theory and Related Topics. Advances in Complex Analysis and Its Applications. Book 3. Springer Science & Business Media.
Li, X. (2011). A Comparison Equality for Rational Functions. Proceedings of the American Mathematical Society. Volume 139, Number 5. Retrieved September 2, 2020 from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.353.2854&rep=rep1&type=pdf
Occhiogrosso, M. (2007). Polar Coordinates and Trigonometric Form: Trigonometry. Milliken Publishing Company.
Robles-Kelly, A. & Hancock, E. (2004). Radiance Function Estimation for Object Classification. In Progress in Pattern Recognition, Image Analysis and Applications. 9th Iberoamerican Congress on Pattern Recognition, CIARP. Springer.
Shiver, J. Polar Equations and Their Graphs. Retrieved September 4, 2020 from: http://jwilson.coe.uga.edu/EMT668/EMAT6680.2003.fall/Shiver/assignment11/PolarGraphs.htm
Image of coordinate grid: Mets501 [CC BY-SA (http://creativecommons.org/licenses/by-sa/3.0/)]


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Stephanie Glen. "Polar Derivative, Coordinates & Function" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/polar-derivative/
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