Distance equations for different cord. systems

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In summary, the conversation is about showing that two equations, ds^2 = dx^2 + dy^2 + dz^2 and ds^2 = dr^2 + r^2(dtheta^2 + sin^2theta*dphi^2), are equivalent. The person asking the question is unsure of how to approach the problem, but understands that they need to use the equations x = r sin theta cos phi, y = r sin theta sin phi, and z = r cos theta. They are then advised to take the partial derivatives of each variable and plug them into the equations to simplify and show their equivalence.
  • #1
aquabug918
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I need to show that

ds^2 = dx^ + dy^2 + dz^2

is the same as

ds^2 = dr^2 + r^2 (dpheta^2 + sin^2pheta*dphi^2)

... I know that I need to use x = r sin pheta cos phi
y = r sin pheta sin phi
z = r cos pheta

I am confused but I think I have to take the derivitive of something. Do I somehow take the partial derivitives and if so of what? Can someone point me in the right direction. Thank you!
 
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  • #2
In general, I believe if you have a function f of several variables x1, ..., xn, then:

[tex]df = \sum _{k=1} ^n\frac{\partial f}{\partial x_k}dx_k[/tex]
 
  • #3
Do Not Post The Same Question Twice
 
  • #4
dx=[itex]\partial x /\partial r dr + \partial x /\partial \theta d\theta + \partial x /\partial \phi d\phi[/itex], and likewise for y and z. Plug in and simplify.
 

FAQ: Distance equations for different cord. systems

What are the different cord systems for measuring distance?

There are three main cord systems for measuring distance - Cartesian coordinates, polar coordinates, and cylindrical coordinates. Each system uses different variables and equations to determine distance.

How is distance calculated in Cartesian coordinates?

In Cartesian coordinates, distance is calculated using the Pythagorean theorem. This means that the distance between two points is equal to the square root of the sum of the squares of the differences between each coordinate.

What is the formula for distance in polar coordinates?

In polar coordinates, distance is calculated using the formula r = √(r₁² + r₂² - 2r₁r₂cos(θ₁ - θ₂)), where r₁ and r₂ are the radii of the two points and θ₁ and θ₂ are the angles from the origin to each point.

How do you find distance in cylindrical coordinates?

In cylindrical coordinates, distance is calculated using the formula d = √(r₁² + r₂² - 2r₁r₂cos(θ₁ - θ₂) + (z₁ - z₂)²), where r₁ and r₂ are the radii of the two points, θ₁ and θ₂ are the angles from the origin to each point, and z₁ and z₂ are the vertical distances from the xy-plane.

Can these distance equations be used in three-dimensional space?

Yes, these distance equations can be used in three-dimensional space as they take into account the coordinates in both the x, y, and z directions. However, the equations may differ slightly depending on the coordinate system being used.

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