Lernen Jacobian Matrix and Change of Variables

Abschnitt 1. Kapitel 8

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The Jacobian matrix is a fundamental concept in multivariate calculus, especially when analyzing how functions transform regions in space. Suppose you have a transformation from variables $x, y$ to new variables $u, v$ defined by two functions:

u = f(x, y)\\ v = g(x, y)

The Jacobian matrix of this transformation is the matrix of all first-order partial derivatives:

J = \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{bmatrix}

The determinant of this matrix, called the Jacobian determinant, measures how areas (or volumes in higher dimensions) are scaled by the transformation at each point. When changing variables in a double integral, the absolute value of the Jacobian determinant adjusts for the stretching or shrinking caused by the transformation.

Example: let's say you have the transformation:

u = x + y\\ v = x - y

The Jacobian matrix is:

J = \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}

The Jacobian determinant is:

|J| = (1)(-1) - (1)(1) = -2

This means that at any point, the transformation scales areas by a factor of 2 (the sign indicates orientation).


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import sympy as sp
import numpy as np

# 1. Define the symbolic variables
x, y = sp.symbols('x y')

# 2. Define our functions as a matrix
funcs = sp.Matrix([x**2 + y, x - y**2])

# 3. Compute the Jacobian symbolically (this replaces the custom jacobian_matrix function)
J_sym = funcs.jacobian([x, y])

# 4. Substitute the values x=1.0, y=2.0
J_eval = J_sym.subs({x: 1.0, y: 2.0})

# Print the results
print("Jacobian matrix at (1, 2):")
print(np.array(J_eval, dtype=float))  # Convert to a numpy array for a standard display format

print("\nJacobian determinant at (1, 2):")
print(float(J_eval.det()))

Aufgabe

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Practice applying your understanding of the Jacobian matrix for a transformation utilizing the sympy library. Given the transformation defined by $h = x \sin(y)$ and $k = y \cos(x)$ , your goal is to compute the exact analytical Jacobian matrix and evaluate it at the point $(1.0, 0.5)$ .

Define the symbolic variables x and y utilizing sympy.symbols().
Define the equations $h$ and $k$ utilizing the sympy sine and cosine functions.
Create a sympy.Matrix representing the system of equations.
Compute the analytical Jacobian matrix utilizing the .jacobian() method.
Substitute the variables x and y with the values 1.0 and 0.5 utilizing the .subs() method.
Convert the evaluated matrix to a numpy array of floats and assign it to the J variable.

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Abschnitt 1. Kapitel 8

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