Example Expressing Vector Fields with Coordinate Systems.doc
1/8
Example: Expressing Vector Fields with Coordinate Systems Consider the vector field: ⎛x A = xz aˆx + ( x 2 + y 2 ) aˆy + ⎜ ⎝z
⎞ ˆ ⎟ az ⎠
Let’s try to accomplish three things: 1. Express A using spherical coordinates and Cartesian base vectors. 2. Express A using Cartesian coordinates and spherical base vectors. 3. Express A using cylindrical coordinates and cylindrical base vectors.
1. The vector field is already expressed with Cartesian base vectors, therefore we only need to change the Cartesian coordinates in each scalar component into spherical coordinates.
Jim Stiles
The Univ. of Kansas
Dept. of EECS
8/23/2005
Example Expressing Vector Fields with Coordinate Systems.doc
2/8
The scalar component of A in the x-direction is:
Ax = xz = ( r sinθ cosφ ) ( r cosθ ) = r 2sinθ cosθ cosφ
The scalar component of A in the y-direction is:
Ay = x 2 + y 2 2
2
= ( r sinθ cosφ ) + ( r sinθ sinφ ) = r 2sin2θ
( cos φ + sin φ ) 2
2
= r 2sin2θ
The scalar component of A in the z-direction is:
x z r sinθ cosφ = r cosθ = tanθ cosφ
Az =
Therefore, the vector field can be expressed using spherical coordinates as: A = r 2sinθ cosθ cosφ aˆx + r 2sin2θ aˆy + tanθ cosφ aˆz
Jim Stiles
The Univ. of Kansas
Dept. of EECS
8/23/2005
Example Expressing Vector Fields with Coordinate Systems.doc
3/8
2. Now, let’s express A using spherical base vectors. We cannot simply change the coordinates of each component. Rather, we must determine new scalar components, since we are using a new set of base vectors. We begin by stating: A = ( A ⋅aˆr ) aˆr + ( A ⋅aˆθ ) aˆθ + ( A ⋅aˆφ ) aˆφ
The scalar component Ar is therefore: ⎛x A ⋅aˆr = xz aˆx ⋅aˆr + ( x 2 + y 2 ) aˆy ⋅aˆr + ⎜ ⎝z
x (x 2 + y 2 ) z x2 +y2 +z2 x2 +y2 x 2z 3 + x 2yz 2 + y 3z − x 3 − xy 2 z x2 +y2 +z2 x2 +y2
And finally, the scalar component Aφ is: ⎛x A ⋅aˆφ = xz aˆx ⋅ aˆφ + ( x 2 + y 2 ) aˆy ⋅ aˆφ + ⎜ ⎝z
⎞ ˆ ˆ ⎟ az ⋅ aφ ⎠ ⎛x ⎞ = xz ( -sinφ ) + ( x 2 + y 2 ) ( cosφ ) + ⎜ ⎟ 0 ⎝z ⎠ −y x = xz + (x 2 + y 2 ) x2 +y2 x2 +y2 =
Jim Stiles
−xyz + x 3 + xy 2
x2 +y2 The Univ. of Kansas
Dept. of EECS
8/23/2005
Example Expressing Vector Fields with Coordinate Systems.doc
5/8
Whew! We’re finished! The vector A is expressed using Cartesian coordinates and spherical base vectors as: ⎛ x 2z + x 2y + y 3 + x A=⎜ ⎜ x2 +y2 +z2 ⎝
⎞ ⎟ aˆr ⎟ ⎠
⎛ x 2z 3 + x 2yz 2 + y 3z − x 3 − xy 2 ⎞ ⎟ aˆθ +⎜ ⎜ z x2 +y2 +z2 x2 +y2 ⎟ ⎝ ⎠ ⎛ −xyz + x 3 + xy 2 ⎞ ⎟ aˆφ +⎜ 2 2 ⎜ ⎟ x +y ⎝ ⎠
3. Now, let’s write A in terms of cylindrical coordinates and cylindrical base vectors (i.e., in terms of the cylindrical coordinate system). A = ( A ⋅ aˆρ ) aˆρ + ( A ⋅ aˆφ ) aˆφ + ( A ⋅ aˆz ) aˆz
First, Aρ is: ⎛x A ⋅aˆρ = xz aˆx ⋅ aˆρ + ( x 2 + y 2 ) aˆy ⋅ aˆρ + ⎜ ⎝z
We can therefore express the vector field A using both cylindrical coordinates and cylindrical base vectors: ⎛ ρ cosφ ⎞ ˆ A = ( ρ cos2φ z + ρ 2sinφ ) aˆρ + ρ cosφ ( ρ − z sinφ ) aˆφ + ⎜ ⎟ az ⎝ z ⎠
Jim Stiles
The Univ. of Kansas
Dept. of EECS
8/23/2005
Example Expressing Vector Fields with Coordinate Systems.doc
7/8
Thus, we have determined three possible ways (and there are many other ways!) to express the vector field A: 1.
A = r 2sinθ cosθ cosφ aˆx + r 2sin2θ aˆy + tanθ cosφ aˆz
2. ⎛ x 2z + x 2y + y 3 + x A=⎜ 2 2 2 ⎜ x y z + + ⎝
⎞ ⎟ aˆr ⎟ ⎠
⎛ x 2z 3 + x 2yz 2 + y 3z − x 3 − xy 2 ⎞ ⎟ aˆθ +⎜ ⎜ z x2 +y2 +z2 x2 +y2 ⎟ ⎝ ⎠ ⎛ −xyz + x 3 + xy 2 ⎞ ⎟ aˆφ +⎜ 2 2 ⎜ ⎟ x +y ⎝ ⎠
3.
⎛ ρ cosφ ⎞ ˆ A = ( ρ cos2φ z + ρ 2sinφ ) aˆρ + ρ cosφ ( ρ − z sinφ ) aˆφ + ⎜ ⎟ az z ⎝ ⎠
Please note: * The three expressions for vector field A provided in this handout each look very different. However, they are just three different methods for describing the same vector field. Any one of the three is correct, and will result in the same result for any physical problem. * We can express a vector field using any set of coordinate variables and any set of base vectors. Jim Stiles
The Univ. of Kansas
Dept. of EECS
8/23/2005
Example Expressing Vector Fields with Coordinate Systems.doc
8/8
* Generally speaking, however, we use one coordinate system to describe a vector field. For example, we use both spherical coordinates and spherical base vectors.
Q: So, which coordinate
system (Cartesian, cylindrical, spherical) should we use ? How can we decide between the three?
A: Ideally, we select that system that most simplifies the mathematics. This depends on the physical problem we are solving. For example, if we are determining the fields resulting from a spherically symmetric charge density, we will find that using the spherical coordinate system will make our analysis the easiest and most straightforward.
