J(x,y)

Definition

The term j(x,y) represents the Jacobian determinant, which is a crucial component in the change of variables when working with double and triple integrals. It measures how much area (in two dimensions) or volume (in three dimensions) is distorted when transforming from one coordinate system to another. This determinant plays a key role in ensuring that integrals are evaluated correctly by adjusting for the stretching or compressing of space caused by the transformation.

5 Must Know Facts For Your Next Test

1. The Jacobian determinant j(x,y) must be computed as the absolute value to ensure that the area or volume is non-negative when transforming coordinates.
2. For a transformation from Cartesian coordinates (x,y) to new coordinates (u,v), j(x,y) is defined as the determinant of the Jacobian matrix formed by the partial derivatives of u and v with respect to x and y.
3. In double integrals, integrating over a transformed region requires multiplying the integrand by j(x,y) to account for the distortion in area.
4. In applications involving polar coordinates, j(x,y) would equal r when transforming from Cartesian coordinates to polar coordinates, affecting how integrals are set up.
5. Understanding how to compute j(x,y) is essential for effectively applying the change of variables theorem in both theoretical and practical problems involving multiple integrals.

Review Questions

• How does the Jacobian determinant j(x,y) affect the evaluation of double integrals during a change of variables?
  • The Jacobian determinant j(x,y) is crucial when evaluating double integrals because it accounts for how areas are transformed under a change of variables. When changing from one set of coordinates to another, j(x,y) adjusts the integral to reflect the actual area being represented in the new coordinate system. Thus, failing to include this determinant can lead to incorrect results when calculating areas or volumes.
• Discuss how you would compute j(x,y) for a transformation from Cartesian coordinates to polar coordinates.
  • To compute j(x,y) when transforming from Cartesian coordinates (x,y) to polar coordinates (r,θ), you first express x and y in terms of r and θ: x = r cos(θ) and y = r sin(θ). Then, you form the Jacobian matrix by taking partial derivatives: ∂(x,y)/∂(r,θ). The determinant of this matrix gives you j(r,θ), which simplifies to r. This value is then used to adjust any double integral evaluated over polar coordinates.
• Evaluate how understanding j(x,y) enhances problem-solving abilities in multivariable calculus applications.
  • Grasping the concept of j(x,y) significantly enhances problem-solving in multivariable calculus by providing a powerful tool for simplifying complex integrals. It allows you to switch between coordinate systems effectively, making it easier to handle integrals over irregular regions or those requiring specific geometric interpretations. This adaptability not only saves time but also aids in visualizing problems more clearly, leading to deeper insights and more accurate solutions across diverse applications.

Related terms

Jacobian Matrix: A matrix consisting of all first-order partial derivatives of a vector-valued function, used to describe how the output changes with respect to changes in input variables.

Transformation: A process that changes the coordinate system or variables used in an integral, which can simplify calculations and provide better insights into the geometry of the problem.

Change of Variables: A mathematical technique that allows for the substitution of one set of variables with another to simplify the evaluation of integrals or to adjust for differences in dimensionality.