Solving for Variables using the Jacobian Transformation

Thread starter robertjford80
Start date May 28, 2012
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Jacobian Transformation

In summary, a Jacobian transformation is a mathematical concept used in multivariate calculus to relate two coordinate systems and describe how changes in one set of variables affect the values of another set of variables. It is defined mathematically by a matrix of partial derivatives and has many applications in various fields, including physics, engineering, computer graphics, and robotics. It is closely related to other mathematical concepts such as the gradient, divergence, and curl, and is essential in understanding change of variables in multiple integrals.

May 28, 2012

robertjford80

Homework Statement

what routine algebra are they talking about. I don't see how they got

x = u/3 - v/3

or

y = 2u/3 +v/3

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May 28, 2012

tiny-tim

Science Advisor

Homework Helper

substitute y = u - x into v, and solve for x

May 28, 2012

robertjford80

thanks I got it. it's amazing how amazing you are.

1. What is a Jacobian transformation?

A Jacobian transformation is a mathematical concept in multivariate calculus that relates two coordinate systems. It is used to describe how changes in one set of variables affect the values of another set of variables. It is commonly used in the study of vector calculus and differential equations.

2. How is a Jacobian transformation defined mathematically?

A Jacobian transformation can be represented by a matrix of partial derivatives, also known as the Jacobian matrix. This matrix is used to calculate the determinant, which represents the change in volume between the two coordinate systems. The Jacobian matrix is also used to determine the inverse transformation.

3. What is the significance of the Jacobian transformation?

The Jacobian transformation is important in many areas of mathematics and science. It is used to simplify complicated calculations in multivariate calculus, and it is also essential in the study of differential equations and optimization problems. It is also used in physics and engineering to describe changes in variables in different coordinate systems.

4. What are some real-world applications of the Jacobian transformation?

The Jacobian transformation has many practical applications in various fields. In physics, it is used to describe the transformation of coordinates in relativity theory. In engineering, it is used in the analysis of fluid flow and heat transfer. It is also used in computer graphics to transform images and in robotics to control robot movement.

5. How does the Jacobian transformation relate to other mathematical concepts?

The Jacobian transformation is closely related to other concepts in mathematics, such as the gradient, divergence, and curl. It is also related to the chain rule and the inverse function theorem. In vector calculus, the Jacobian matrix is used to calculate the Jacobian determinant, which is equivalent to the scalar triple product. Additionally, the Jacobian transformation is essential in understanding the change of variables in multiple integrals.