Lecture XVII Curvilinear Coordinates; Change of Variables Flipbook PDF

Lecture XVII Curvilinear Coordinates; Change of Variables As we saw in lecture 16, in E2 we can use the polar coordinate

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Lecture XVII Curvilinear Coordinates; Change of Variables

As we saw in lecture 16, in E2 we can use the polar coordinates system. � The coordinates In this system, we have a fixed point O and a fixed ray Ox. of a point P are given by r, the distance from P to O, and θ the angle made � and Ox, � as measured going counterclockwise from Ox � to OP � . We can by OP change the system of coordiantes from polar to Cartesian through a system of equations: x = r cos θ,

y = r sin θ.

This is called the defining system. To go from the Cartesian system to the polar one, we use the inverse defining system: � y r = x2 + y 2 , θ = arctan , x for (x, y) in the first quadrant. Generally, we can introduce new non­Cartesian coordinates u, v by writing x, y as functions of these new coordiantes: x = g(u, v),

y = h(u, v).

These equations form the defining system for the new coordiantes. They can be summarized by writting the position vector � R(u, v) = xˆi + yjˆ = g(u, v)ˆi + h(u, v)ˆj. In working with curvilinear coordinates, it is useful to introduce unit coordinate vectors u ˆP , vˆP . To find u ˆP , we take � ∂g ˆ ∂h ˆ ∂R |P = |P i + |P j, ∂u ∂u ∂u and we form the unit vector u ˆP = � �

∂R ∂R ∂v |P /| ∂v |P |.

� � ∂R ∂R ∂u |P /| ∂u |P | .

Similarly, we define vˆP =

For example, recall that for polar coordinates, the defining system

is x = r cos θ, y = r sin θ. The position vector is: � = r cos θˆi + r cos θˆj, R 1

and the unit coordinate vectors are: � ∂R = cos θˆi + sin θˆj = r, ˆ ∂r � ∂R ˆ = − sin θˆi + cos θˆj = θ. ∂θ In the uv plane, consider the rectangle determined by the vectors uΔu ˆ and vΔv ˆ with point P as their tail, where Δu and Δv are two small positive quantities. Now consider the point P � correspoding to P and the region corresponding to the rectangle in the xy plane. The area of this region is given by � � � � � � ∂R � ∂R � � �� ∂R ∂R � � � � � | Δu × |P Δv � = � | × |P � ΔuΔv � � � ∂u P � ∂u P ∂v ∂v � This vector product can be easily computed, since � ∂g ˆ ∂h ˆ ∂R = i+ j ∂u ∂u ∂u Hence

� � � � � ∂R ∂R × = �� ∂u ∂v �

and

∂x ∂u ∂y ∂u

� ∂R ∂g ˆ ∂h ˆ = i+ j. ∂v ∂v ∂v

∂x ∂v ∂y ∂v

� � � ∂(x, y) �= � ∂(u, v) . �

This expression is called the Jacobian of x and y with respect to u and v. For polar coordinates, the Jacobian is � � ∂(x, y) �� cos θ −r sin θ = ∂(u, v) �� sin θ r cos θ

� � � � = r cos2 θ + r sin2 θ = r. � �

When switching from the Cartesian system to a curvilinear one, u, v might be given as functions of x, y, so the following equality is useful: � ∂(u, v) ∂(x, y) =1 . ∂(x, y) ∂(u, v) The notion of Jacobian can be extended to E3 in the following manner. If x, y, z are functions of u, v, w, the Jacobian is � � ∂x � ∂u � ∂(x, y, z) = � ∂y ∂(u, v, w) �� ∂u � ∂z ∂u

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∂x ∂v ∂y ∂v ∂z ∂v

∂x ∂w ∂y ∂w ∂z ∂w

� � � � �. � � �

For spherical coordinates ρ, ϕ, θ, the Jacobian is: ∂(x, y, z) = r2 sin ϕ. ∂(ρ, ϕ, θ) Jacobians are particularly useful when we compute integrals, because we can change variables in the following way: � � � � � � � ∂(x, y) � � dudv. f (x, y)dxdy = f (g(u, v), h(u, v)) �� ∂(u, v) � ˆ R R

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