For triple integrals, there is a similar formula for change of variables. Suppose $ x = x(s, t, u), \, y = y(s, t, u), \, z = z(s, t, u) $ define a change of variables from a region $ S $ in $ stu $-space to a region $ W $ in $ xyz $-space. Then the Jacobian of the change of variables is given by the determinant:\[\frac{\partial(x, y, z)}{\partial(s, t, u)} = \begin{vmatrix} \frac{\partial x}{\partial s} & \frac{\partial x}{\partial t} & \frac{\partial x}{\partial u} \\ \frac{\partial y}{\partial s} & \frac{\partial y}{\partial t} & \frac{\partial y}{\partial u} \\ \frac{\partial z}{\partial s} & \frac{\partial z}{\partial t} & \frac{\partial z}{\partial u} \end{vmatrix}.\]Compute the Jacobian for the change of variables into spherical coordinates where $ x = \rho \sin(\phi) \cos(\theta), \, y = \rho \sin(\phi) \sin(\theta), \, z = \rho \cos(\phi) $. **You must simplify completely. The final answer has a very simple form.**

Given: x=x(s,t,u) , y=y(s,t,u), z=(s,t,u) define a change of variable from a region S in stu-space…

For triple integrals, there is similar formula for change of variables. Suppose = x(s,t, u), y = y(s,t, u), z = 5 in stu-space to a region W in xyz-space. Then the Jacobian of the is change of rariables in given by the determinant: z(s, t, u) define a change of variables from a region ds Ət ди a(x, y, z) dy dy ds dy ди a(s, t, u) dz ds dz dz du Compute the Jacobian for the change of variables into spherical coordinates where : = psin(ø) cos(), y = psin(ø) sin(0), z = pletely. The final answer has a very simple form. p cos(ø). You must simplify com-

For triple integrals, there is similar formula for change of variables. Suppose = x(s,t, u), y = y(s,t, u), z = 5 in stu-space to a region W in xyz-space. Then the Jacobian of the is change of rariables in given by the determinant: z(s, t, u) define a change of variables from a region ds Ət ди a(x, y, z) dy dy ds dy ди a(s, t, u) dz ds dz dz du Compute the Jacobian for the change of variables into spherical coordinates where : = psin(ø) cos(), y = psin(ø) sin(0), z = pletely. The final answer has a very simple form. p cos(ø). You must simplify com-

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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For triple integrals, there is a similar formula for change of variables. Suppose $ x = x(s, t, u), \, y = y(s, t, u), \, z = z(s, t, u) $ define a change of variables from a region $ S $ in $ stu $-space to a region $ W $ in $ xyz $-space. Then the Jacobian of the change of variables is given by the determinant:

\[
\frac{\partial(x, y, z)}{\partial(s, t, u)} =
\begin{vmatrix}
\frac{\partial x}{\partial s} & \frac{\partial x}{\partial t} & \frac{\partial x}{\partial u} \\
\frac{\partial y}{\partial s} & \frac{\partial y}{\partial t} & \frac{\partial y}{\partial u} \\
\frac{\partial z}{\partial s} & \frac{\partial z}{\partial t} & \frac{\partial z}{\partial u}
\end{vmatrix}.
\]

Compute the Jacobian for the change of variables into spherical coordinates where $ x = \rho \sin(\phi) \cos(\theta), \, y = \rho \sin(\phi) \sin(\theta), \, z = \rho \cos(\phi) $. **You must simplify completely. The final answer has a very simple form.**

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