Proving r^2 = x^2 + y^2 in Polar Coordinates

  • Thread starter mrcleanhands
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In summary, the conversation discusses a multi-variable calculus assignment where the task is to show that r^2 = x^2 + y^2 and theta = atan(y/x) using the given equation z = f(x,y) and the standard polar coordinate equations. The conversation also mentions using the Pythagorean Theorem to prove this, but ultimately concludes that it is not necessary.
  • #1
mrcleanhands

Homework Statement


Consider z=f(x,y), where x=rcosθ and y=rsinθ
(This is a multi part multi variable calculus assignment question). I've just derived dz/dr and now I'm asked... to show that [itex]r^{2}=x^{2}+y^{2},
\theta=a\tan(\frac{y}{x})[/itex]

Homework Equations


The Attempt at a Solution



I've drawn up a triangle with r as the hypotenuse, x on the x axis, y on the y axis.

and then x=rcosθ=r * x/r = x by trig laws and then [itex]r^{2}=x^{2}+y^{2}[/itex] by pythagoras theorem etc


this seems too easy. is there some other way to show this?
 
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  • #2
Not unless you want to prove the Pythagorean Theorem itself.
 
  • #3
mrcleanhands said:

Homework Statement


Consider z=f(x,y), where x=rcosθ and y=rsinθ
(This is a multi part multi variable calculus assignment question). I've just derived dz/dr and now I'm asked... to show that [itex]r^{2}=x^{2}+y^{2},
\theta=a\tan(\frac{y}{x})[/itex]

That doesn't have anything to do with ##z = f(x,y)##. It's just standard polar coordinate equations.$$
x^2 + y^2 = r^2\cos^2\theta + r^2\sin^2\theta = r^2$$ $$
\frac y x = \frac{r\sin\theta}{r\cos\theta}=\tan\theta$$
 
  • #4
yeah that's why I was confused. I don't get how I'm showing that by just plugging it into equations and referring to the polar equations.
 

1. What does the equation r^2 = x^2 + y^2 represent?

The equation r^2 = x^2 + y^2 represents the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (r) is equal to the sum of the squares of the other two sides (x and y).

2. How can this equation be used in science?

This equation has many applications in science, particularly in fields such as physics and engineering. It can be used to calculate the distance between two points, such as the location of an object in space. It is also used to calculate the magnitude of a vector in physics.

3. Can r^2 = x^2 + y^2 be applied to non-right triangles?

No, this equation can only be applied to right triangles, as it is based on the Pythagorean theorem which specifically applies to right triangles.

4. How is this equation related to the Cartesian coordinate system?

The Cartesian coordinate system is based on the idea of plotting points on a grid using x and y coordinates. This equation relates to the Cartesian coordinate system as it is used to calculate the distance between two points on the grid.

5. Are there any real-world examples of r^2 = x^2 + y^2?

Yes, there are many real-world examples of this equation. For instance, it can be used to calculate the distance between two cities on a map, or to determine the length of a diagonal fence in a rectangular yard. It is also used in navigation and GPS systems to calculate distances between locations.

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