jacobian

Meanings

• noun

  In mathematics, the Jacobian matrix is a square matrix of first-order partial derivatives of a function.

  - "The Jacobian matrix of a function f(x, y) is given by [[df/dx, df/dy], [dg/dx, dg/dy]]."
  - "The determinant of the Jacobian matrix of a transformation is used to calculate the Jacobian of the transformation."

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Variants

List of all variants of jacobian that leads to same result
Jacobian , Jacobians

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Etymology

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Named after the German mathematician Carl Gustav Jacob Jacobi (1804-1851).

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Trivia

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1. Carl Gustav Jacob Jacobi was also known for his work on elliptic functions and number theory.

2. The Jacobian matrix is a generalization of the gradient vector in the multivariable case.

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Related Concepts

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1. Jacobian transformation: A transformation that preserves the volume element in the change of variables in multivariable calculus.

2. Jacobian determinant: The determinant of the Jacobian matrix, which can be used to calculate the Jacobian of a transformation.

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Culture

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The Jacobian matrix is a fundamental concept in mathematics, particularly in the fields of calculus and linear algebra. It is used extensively in physics, engineering, and computer science to understand the behavior of functions and transformations.

How to Memorize "jacobian"

1. visualize
   

   - Imagine a 2x2 matrix with two rows and two columns. Each entry represents the first-order partial derivative of a function with respect to a specific variable.
   - Visualize the transformation of a function from one coordinate system to another and the corresponding Jacobian matrix.

2. associate
   

   - Associate the Jacobian matrix with the concept of a transformation and its effect on the volume element.
   - Associate the determinant of the Jacobian matrix with the concept of the Jacobian of a transformation.

3. mnemonics
   

   - Jacobian: Just Another Cool Basis for Integration and Approximation

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