∂(x,y)/∂(u,v)

Definition

The partial derivative of the variables (x,y) with respect to the variables (u,v) is a mathematical expression that describes the rate of change of the (x,y) coordinates with respect to changes in the (u,v) coordinates. This concept is particularly important in the context of change of variables in multiple integrals.

5 Must Know Facts For Your Next Test

1. The partial derivative ∂(x,y)/∂(u,v) is a crucial component in the formula for changing variables in a multiple integral.
2. The value of ∂(x,y)/∂(u,v) represents the scaling factor that relates infinitesimal changes in the (u,v) coordinates to the corresponding changes in the (x,y) coordinates.
3. The Jacobian of the transformation is equal to the absolute value of ∂(x,y)/∂(u,v).
4. The change of variables formula in multiple integrals involves multiplying the original integrand by the absolute value of the Jacobian, \left|\frac{\partial(x,y)}{\partial(u,v)}\right|.
5. The sign of ∂(x,y)/∂(u,v) determines the orientation of the transformation, which is important for correctly applying the change of variables formula.

Review Questions

• Explain the role of ∂(x,y)/∂(u,v) in the context of change of variables in multiple integrals.
  • The partial derivative ∂(x,y)/∂(u,v) is a key component in the change of variables formula for multiple integrals. It represents the scaling factor that relates infinitesimal changes in the (u,v) coordinates to the corresponding changes in the (x,y) coordinates. The absolute value of this partial derivative, known as the Jacobian, is used to multiply the original integrand when transforming the integral to the new (u,v) coordinates. The sign of ∂(x,y)/∂(u,v) also determines the orientation of the transformation, which is important for correctly applying the change of variables formula.
• Describe how the Jacobian is related to the partial derivative ∂(x,y)/∂(u,v).
  • The Jacobian of the transformation between the (x,y) and (u,v) coordinates is defined as the determinant of the matrix of partial derivatives, \begin{vmatrix}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\end{vmatrix}. This determinant is equal to the absolute value of the partial derivative ∂(x,y)/∂(u,v). The Jacobian represents the scaling factor that relates infinitesimal changes in the (u,v) coordinates to the corresponding changes in the (x,y) coordinates, which is a crucial component in the change of variables formula for multiple integrals.
• Analyze the importance of the sign of ∂(x,y)/∂(u,v) in the context of change of variables in multiple integrals.
  • The sign of the partial derivative ∂(x,y)/∂(u,v) is important in the context of change of variables in multiple integrals because it determines the orientation of the transformation between the (x,y) and (u,v) coordinates. If the sign is positive, the transformation preserves the orientation, meaning that the new coordinates (u,v) have the same orientation as the original coordinates (x,y). If the sign is negative, the transformation reverses the orientation. Knowing the orientation of the transformation is crucial for correctly applying the change of variables formula, which involves multiplying the original integrand by the absolute value of the Jacobian, \left|\frac{\partial(x,y)}{\partial(u,v)}\right|.

Related terms

Jacobian:

The Jacobian is the determinant of the matrix of partial derivatives that describes the transformation between the original variables and the new variables in a change of variables.

Change of Variables:

The process of transforming the variables in a multiple integral to simplify the integration or better fit the geometry of the region of integration.

Partial Derivative:

A partial derivative is the derivative of a function with respect to one of its variables, treating the other variables as constants.