What are polar coordinates? What equations relate polar coordi-nates to Cartesian coordinates? Why might you want to change from one coordinate system to the other?

djeljenike 2021-02-27 Answered
What are polar coordinates? What equations relate polar coordi-nates to Cartesian coordinates? Why might you want to change from one coordinate system to the other?
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Expert Answer

Margot Mill
Answered 2021-02-28 Author has 106 answers
Step 1
Consider an origin as O and an initial ray from the origin. The positive x-axis is usually considered as initial ray.
Polar coordinate is a point P (r,θ). Here, r is the directed distance from origin O to point P and the angle theta is the directed angle from initial ray to ray OP.
Write the expression for relation between Cartesian and polar equations as follows:
x=rcosθ
y=rsinθ
r2=x2+y2
tanθ=yx
Here, (x, y) is the Cartesian coordinate.
Step 2
Certian coordinate systems are quite complex to analyse in a particular coordinate system.
For example, polar coordinate system is very useful for many multiple integral systems such as description of paths of planets and satellites. But, it more complex to describe the planets and satellites by using Cartesian coordinate system.
Therefore, the Polar coordinates are changed to Cartesian coordinates and vice versa for the analysis of various systems with ease.
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Consider the following.

z = x2 + y2,   x = 5s + 6t,   y = s + t

Find 

z
s

 and 

z
t

 by using the chain rule. (Enter your answers in terms of s and t.)

z
s

=

52s+62t

 

z
t

=

62s+74t

 

Find 

z
s

 and 

z
t

 by first substituting the expressions for x and y to write z as a function of s and t. (Enter your answers in terms of s and t.)

z
s

=

 

 

z
t

=

 

Do your answers for 

z
s

 agree?

YesNo    

Do your answers for 

z
t

 agree?

YesNo