Confusion about variables in polar coordinates

In summary, "Confusion about variables in polar coordinates" discusses the common misunderstandings related to the use of polar coordinates in mathematics and physics. It highlights how the representation of points in polar coordinates, defined by a radius and an angle, can lead to ambiguity in calculations, especially when converting to Cartesian coordinates or interpreting the meaning of the variables. The article emphasizes the importance of clarity in notation and understanding the relationships between the coordinate systems to avoid errors in analysis and problem-solving.
  • #1
sdfsfasdfasf
75
12
Homework Statement
x
Relevant Equations
x
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My confusion refers to this question above.

If I were to ask you, what is the equation of the radial line, what would you say? I know that the general equation the radial line with cartesian gradient of m has an equation of θ = arctan(m). Clearly here the angle between the radial line and initial line is θ, therefore the equation is θ = θ? That can't be right, do we call the angle θ a different name, like α? Then we'd have θ = α, which seems better. How does this get around the problem?
Things like this don't really confuse me usually, can someone help me out here? Is it bad writing from the author?

Thank you for reading
 
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  • #2
sdfsfasdfasf said:
Homework Statement: x
Relevant Equations: x

If I were to ask you, what is the equation of the radial line, what would you say?
I'd say, θ = const.
 
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Likes Mark44
  • #3
Just to add: This is the polar coordinate equivalent of writing the equation for a line parallel to the x-axis in Cartesian coordinates, which is just ##y =## constant.
 
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FAQ: Confusion about variables in polar coordinates

What are polar coordinates?

Polar coordinates are a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis). The coordinates are typically represented as (r, θ), where r is the radial distance and θ is the angular coordinate.

How do I convert between polar and Cartesian coordinates?

To convert from polar coordinates (r, θ) to Cartesian coordinates (x, y), you can use the formulas: x = r * cos(θ) and y = r * sin(θ). Conversely, to convert from Cartesian coordinates (x, y) to polar coordinates (r, θ), use r = √(x² + y²) and θ = atan2(y, x), where atan2 is the two-argument arctangent function that accounts for the sign of both x and y to determine the correct quadrant of θ.

What is the significance of the angle in polar coordinates?

The angle θ in polar coordinates indicates the direction of the point relative to the positive x-axis. It is usually measured in radians or degrees and can affect the representation of the same point when expressed in different quadrants or when the angle is adjusted by adding or subtracting multiples of 2π (360 degrees).

Why do polar coordinates sometimes lead to confusion?

Confusion often arises in polar coordinates due to the periodic nature of the angle θ. A single point can be represented by multiple (r, θ) pairs, such as (r, θ) and (r, θ + 2πn) for any integer n. Additionally, negative values of r can represent the same point as positive values but with a different angle (θ + π). This can lead to misunderstandings when interpreting or visualizing the coordinates.

How do I graph functions in polar coordinates?

To graph functions in polar coordinates, you typically express them in the form r = f(θ), where r is a function of the angle θ. You then calculate the value of r for various angles θ within a specified range and plot the corresponding points on a polar grid. Connecting these points provides the visual representation of the function. Understanding the periodic nature of the angle is crucial for accurately capturing the shape of the graph.

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