The change of variables theorem is a powerful tool for simplifying complex integrals. It allows us to transform double and triple integrals into new coordinate systems, making them easier to evaluate. This process involves substituting variables and using the Jacobian determinant.

By applying this theorem, we can tackle integrals that would be challenging in their original form. It's especially useful for problems involving unusual shapes or complicated functions, as it can transform them into more manageable forms.

Change of Variables Theorem

Applying the Change of Variables Theorem

• Change of variables theorem allows for simplifying the evaluation of double and triple integrals by transforming the integral to a new coordinate system
• Double integrals can be transformed using the change of variables theorem
  • Involves substituting new variables $(u,v)$ for the original variables $(x,y)$
  • Requires the use of the Jacobian determinant to account for the change in area elements
• Triple integrals can also be transformed using the change of variables theorem
  • Involves substituting new variables $(u,v,w)$ for the original variables $(x,y,z)$
  • Requires the use of the Jacobian determinant to account for the change in volume elements
• Substitution method is used to simplify the integrand and/or the region of integration
  • Choose new variables that simplify the integrand and/or transform the region into a more manageable shape (rectangles, circles, spheres)
  • Example: Transforming a double integral over an elliptical region to a circular region using polar coordinates $(r,\theta)$

Key Components of the Change of Variables Theorem

• The change of variables theorem states that for a double integral:
  • $\iint_{R} f(x,y) dA = \iint_{S} f(x(u,v),y(u,v)) \left| \frac{\partial(x,y)}{\partial(u,v)} \right| dA$
  • $R$ is the original region of integration, $S$ is the transformed region
• For a triple integral, the change of variables theorem states:
  • $\iiint_{R} f(x,y,z) dV = \iiint_{S} f(x(u,v,w),y(u,v,w),z(u,v,w)) \left| \frac{\partial(x,y,z)}{\partial(u,v,w)} \right| dV$
• The term $\left| \frac{\partial(x,y)}{\partial(u,v)} \right|$ is the absolute value of the Jacobian determinant for double integrals
• The term $\left| \frac{\partial(x,y,z)}{\partial(u,v,w)} \right|$ is the absolute value of the Jacobian determinant for triple integrals

Transformation and Jacobian

Transforming Regions and Integration Limits

• Transformation of regions involves mapping the original region of integration to a new region in the transformed coordinate system
  • The shape of the region may change depending on the chosen transformation
  • Example: A rectangular region in Cartesian coordinates may transform into a polar rectangle in polar coordinates
• Integration limits must be adjusted to correspond to the new region in the transformed coordinate system
  • Determine the equations of the boundaries in terms of the new variables
  • Example: For a double integral over a circular region, the limits in polar coordinates would be $0 \leq r \leq a$ and $0 \leq \theta \leq 2\pi$

Jacobian Determinant and Its Role

• The absolute value of the Jacobian determinant, denoted as $\left| \frac{\partial(x,y)}{\partial(u,v)} \right|$ for double integrals or $\left| \frac{\partial(x,y,z)}{\partial(u,v,w)} \right|$ for triple integrals, is a scaling factor that relates the area or volume elements between the original and transformed coordinate systems
• The Jacobian determinant accounts for the change in the size of the area or volume elements when transforming the integral
  • It is the determinant of the Jacobian matrix, which consists of the partial derivatives of the original variables with respect to the new variables
• Area elements $(dA)$ in double integrals are transformed using the Jacobian determinant
  • Example: In polar coordinates, $dA = r , dr , d\theta$, where $r$ is the Jacobian determinant
• Volume elements $(dV)$ in triple integrals are transformed using the Jacobian determinant
  • Example: In spherical coordinates, $dV = \rho^2 \sin\phi , d\rho , d\phi , d\theta$, where $\rho^2 \sin\phi$ is the Jacobian determinant