Welcome to four examples
of converting a rectangular equation
to a cylindrical equation.
We're asked to find an equation in cylindrical coordinates
for the rectangular equation z
equals x squared plus y squared minus two
which is graphed below.
Notice how the graph is a paraboloid.
For a quick review, in three dimensions
when we have a point in rectangular form,
it's in the form, x comma y comma z in cylindrical form
but is just an extension of polar coordinates
in two dimensions to three dimensions.
Our point is in the form, r comma theta comma z
which are pictured here in the three dimensional graph.
Comparing these coordinates tells us
to write a rectangular equation in cylindrical form,
we must perform a substitution for x and y
so that the new equation in cylindrical form
only contains the variables r, theta and z.
To perform the conversions,
we use the equations shown below
which relate x, y, z, r and theta.
So going back to our equation,
z equals x squared plus y squared minus two,
we must perform a substitution for both x and y.
Looking at our equations, notice how
r squared equals x squared plus y squared.
Using this equation to perform a substitution
for x squared plus y squared
gives us the equation, z equals r squared minus two.
Notice how this equation only contains z and r
and therefore it is now in the cylindrical form
and it is common when writing an equation
in cylindrical form to either solve the equation for r or z.
Let's look at another example.
We were given the equation,
y squared equals 10 minus z squared.
Again, it's okay to have the variable z in the equation
but not the variable y
because y is equal to r sine theta.
We use this equation to perform a substitution
for y squared.
y squared is equal to the square of r sine theta.
Our equation is now the square of r sine theta
equals 10 minus z squared.
Notice how the equation only contains the variables,
r, theta and z and therefore it is in cylindrical form
but let's go ahead and simplify it further.
Let's square the left side
which would give us r squared sine squared theta
then we have equals 10 minus z squared.
Let's go ahead and solve this for z squared
by adding z squared to both sides
and also subtracting r squared sine squared theta
from both sides.
This would give us the equation, z squared equals 10
minus r squared sine squared theta.
Either of these last two equations
would be common forms for the equation in cylindrical form.
For our third example,
we are given the rectangular equation x equals y squared
which is graphed below.
Notice how we have a parabolic cylinder.
For this equation and we need to perform a substitution
for both x and y,
x equals r cosine theta,
y equals r sine theta.
Performing this substitution gives us r cosine theta
equals y squared is going to be r squared sine squared theta
which we saw from the previous example.
Notice how this equation only contains r and theta
therefore it is in cylindrical form
but let's go ahead and solve this for r.
To solve this for r, let's first divide both sides by r.
Simplifying, r divided by r simplifies to one.
We have cosine theta
equals on the right side, r squared divided by r
simplifies to one factor of r.
We have r sine squared theta.
Now, to solve for r, we'll divide both sides
by sine squared theta.
Simplifying on the right.
Sine squared theta divided by sine squared theta
simplifies to one.
Let's go ahead and change the order of the equation
and write this as r equals cosine theta
divided by sine squared theta.
This might be one common form to leave the equation in
using cylindrical coordinates or cylindrical form.
Let's go ahead and write this down
but let's also show how we can change the formulas
so there's no fraction involved.
We can also write this as cosine theta
divided by sine theta times one over sine theta.
Notice how we still have cosine theta
divided by sine squared theta
but in this form, cosine theta divided by sine theta
is equal to cotangent theta
and one over sine theta is equal to cosecant theta.
So we could also write this as r equals cotangent theta
times cosecant theta.
Either of these two forms would be common forms
for the equation in cylindrical form.
Let's look at one more example.
Here, we're given the equation x squared plus y squared
equals six y which is graphed below.
Notice we have a cylinder.
To write using cylindrical coordinates,
we need to perform a substitution
for x squared plus y squared and y.
x squared plus y squared equals r squared
and y equals r sine theta
which gives us the equation, r squared
equals six r sine theta.
Because the equation only contains r and theta now,
it is in cylindrical form
but let's go ahead and solve this for r
by dividing both sides by r.
On the left, r squared divided by r is equal to r.
On the right, r divided by r simplifies to one
giving us the equation r equals six sine theta.
This is the most common form for the cylindrical equation.
I hope you found this helpful.