17Calculus - Cylindrical Coordinates

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This page covers the basics of cylindrical coordinates and reference equations that use cylindrical coordinates for multi-variable calculus.

When you studied polar coordinates, you learned that you could take an area in the plane in rectangular coordinates and use polar coordinates to describe the same area. To do this, you used the equations $$x=r\cos(\theta)$$ and $$y=r\sin(\theta)$$. These equations convert the equations for an area in the plane from rectangular coordinates $$(x,y)$$ to polar coordinates $$(r,\theta)$$.

Table 1 $$x=r\cos(\theta)$$ $$y=r\sin(\theta)$$ $$z=z$$ $$r^2=x^2+y^2$$

However, now we have three dimensions. In rectangular coordinates, we have $$(x,y,z)$$. One of the nice things about cylindrical coordinates is that we use the same equations on x and y that we used for polar coordinates to get r and $$\theta$$ and to go to cylindrical coordinates z does not change. Another way of looking at it is that we take polar coordinates $$(r,\theta)$$ and slap on the rectangular coordinate z to the end to get $$(r,\theta,z)$$ and call this cylindrical coordinates. So the cylindrical coordinates conversion equations are given in Table 1 and Figure 1 shows this relationship.

Figure 1

[source: Paul's Online Notes]

Figure 2

Comment About Notation - Figure 2 shows another way to describe cylindrical coordinates. Some books, instructors, videos and sites use $$(\rho,\phi,z)$$ to describe the same point as $$(r,\theta,z)$$. We choose to use $$(r,\theta,z)$$ for cylindrical coordinates since the cylindrical coordinate system is so closely related to two-dimensional polar coordinates usually described at $$(r,\theta)$$. As always, check with your instructor to see what they expect you to use.
Also in Figure 2, they are calling A the polar axis (positive x-axis) and L is the positive z-axis.

Del Operator and Cylindrical Coordinates

Michel vanBiezen - Del Operator in Cylindrical Coordinate [3mins-50secs]

video by Michel vanBiezen

You CAN Ace Calculus

Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

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