MATH 200 WEEK 8 - FRIDAY SPHERICAL AND CYLINDRICAL COORDINATES
MATH 200 GOALS ▸ Be able to convert between the three different coordinate systems in 3-Space: rectangular, cylindrical, spherical ▸ Develop a sense of which surfaces are best represented by which coordinate systems
MATH 200 CYLINDRICAL COORDINATES ▸ Cylindrical coordinates are z basically polar coordinates θ r plus z ▸ Coordinates: (r, θ ,z) ▸ x = rcos θ ▸ y = rsin θ y JUST LIKE 2D POLAR θ r ▸ z = z x ▸ r 2 = x 2 + y 2 ▸ tan θ = y/x
MATH 200 SURFACES ▸ Let’s look at the types of surfaces we get when we set polar coordinates equal to constants. ▸ Consider the surface r = 1 ▸ This is the collection of all points 1 unit from the z- axis ▸ Or, using our transformation equations, it’s the same as the surface x 2 +y 2 =1
MATH 200 ▸ How about θ =c? ▸ This is the set of all points for which the θ component is fixed, but r and z can be anything. ▸ Or, since tan θ = c, we have y/x = c ▸ y = cx is a plane θ
MATH 200 SPHERICAL COORDINATES ▸ Coordinates: ( ρ , θ , φ ) z θ ▸ ρ : distance from origin to point ▸ θ : the usual θ (measured ρ off of positive x-axis) φ y ▸ φ : angle measured from θ positive z-axis x
MATH 200 CONVERTING ▸ Let’s start with ρ : z θ ▸ From the distance formula/Pythagorus we get ρ 2 =x 2 +y 2 +z 2 z ρ ▸ We already know that φ tan θ =y/x y θ ▸ Lastly, since z = ρ cos φ , we have x For φ , z is the adjacent side z cos φ = � x 2 + y 2 + z 2
MATH 200 ▸ Going the other way r is the opposite side to φ around is a little trickier… z ▸ From cylindrical/polar, we r θ have � x = r cos θ ρ y = r sin θ φ ▸ Notice that r = ρ sin φ . So, y θ x = ρ sin φ cos θ x y = ρ sin φ sin θ z = ρ cos φ
MATH 200 SURFACES IN SPHERICAL ▸ Let’s start with ρ =constant ▸ What does ρ =2 look like? ▸ It’s all points 2 units from the origin ▸ Also, if ρ =2, then ρ 2 =4. So, x 2 +y 2 +z 2 =4 ▸ It’s a sphere!
MATH 200 ▸ How about φ =constant? � x 2 + y 2 + z 2 = 2 z x 2 + y 2 + z 2 = 4 z 2 ▸ Let φ = π /3. x 2 + y 2 = 3 z 2 ▸ From the conversion z 2 = 1 3 x 2 + 1 formula we have 3 y 2 z cos π 3 = ▸ Recall: z 2 =x 2 +y 2 is a double x 2 + y 2 + z 2 � cone 1 z 2 = x 2 + y 2 + z 2 � ▸ Multiplying the right-hand side by 1/3 just stretches it ▸ Let’s simplify some
MATH 200 ▸ For spherical coordinates, we restrict ρ and φ ▸ ρ≥ 0 and 0 ≤φ≤π ▸ So, φ = π /3 is just the top of the cone
MATH 200 EXAMPLE 1: CONVERTING POINTS ▸ Consider the point ( ρ , θ , φ ) = (5, π /3, 2 π /3) ▸ Convert this point to rectangular coordinates ▸ Convert this point to cylindrical coordinates ▸ Rectangular � � 1 � √ � 3 � � � x = 5 x = 5 sin 2 π x = ρ sin φ cos θ 3 cos π � 2 2 3 � � � � � � � √ � � √ � y = 5 sin 2 π y = ρ sin φ sin θ = 3 sin π = 3 3 ⇒ ⇒ y = 5 3 2 2 � � � z = 5 cos 2 π z = ρ cos φ � � � � − 1 � z = 5 � 3 2 ▸ In rectangular coordinates, we have √ � � 5 4 , 15 3 4 , − 5 ( x, y, z ) = 2
MATH 200 ▸ Polar: ▸ We already have z and θ : √ � � r 2 = x 2 + y 2 5 3 3 , − 5 2 , π ( r, θ , z ) = 2 � 2 � � 2 √ � 15 5 3 r 2 = + 4 4 r 2 = 75 16 + 225 16 r 2 = 300 16 √ r = 10 3 4 √ r = 5 3 2
MATH 200 φ ρ θ
MATH 200 EXAMPLE 2: CONVERTING SURFACES ▸ Express the surface x 2 +y 2 +z 2 =3z in both cylindrical and spherical coordinates ▸ Cylindrical ▸ Using the fact that r 2 =x 2 +y 2 , we have r 2 +z 2 =3z ▸ Spherical ▸ Using the facts that ρ 2 =x 2 +y 2 +z 2 and z = ρ cos φ , we get that ρ 2 =3 ρ cos φ ▸ More simply, ρ =3cos φ
MATH 200 EXAMPLE 3: CONVERTING MORE SURFACES ▸ Express the surface ρ =3sec φ in both rectangular and cylindrical coordinates ▸ We can rewrite the equation as ρ cos φ =3 ▸ This is just z = 3 (a plane) ▸ Conveniently, this is exactly the same in cylindrical!
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