Pullback Measure Calculator

Advanced Pullback Measure Calculator

Source dimension

Target measure model

Constant density

1D scale a

Used in T(x) = ax + b.

1D shift b

Source interval start

Source interval end

Sample x value

1D integration samples

Matrix A11

Matrix A12

Matrix A21

Matrix A22

Translation c1

Translation c2

u minimum

u maximum

v minimum

v maximum

Sample u

Sample v

2D grid points per side

Gaussian mean y1

Gaussian sd y1

Gaussian mean y2

Gaussian sd y2

Target y1 minimum

Target y1 maximum

Target y2 minimum

Target y2 maximum

Formula Used

General density formula:

T^*μ(E) = ∫_E ρ(T(x)) |det DT(x)| dx

For one dimension, |det DT(x)| becomes |T′(x)|.

For an affine map T(x) = Ax + c and Lebesgue measure, the result is |det A| × volume(E).

How to Use This Calculator

Select a 1D interval or 2D rectangle.
Choose the target measure model.
Enter the transformation parameters.
Enter source domain limits.
Add Gaussian or target box values when required.
Press the calculate button.
Review the result above the form.
Export the result as CSV or PDF.

Example Data Table

Case	Transformation	Source Domain	Target Measure	Jacobian	Pullback Measure
1D stretch	T(x) = 2x + 1	[0, 3]	Lebesgue	2	6
1D constant density	T(x) = 3x	[1, 4]	ρ = 0.5	3	4.5
2D scaling	A = [[2,0],[0,3]]	[0,2] × [0,2]	Lebesgue	6	24
2D density	A = [[1,1],[0,2]]	[0,1] × [0,1]	Gaussian	2	Numerical estimate

Understanding Pullback Measure

What It Means

A pullback measure describes how a measure on a target space is read on a source space. It uses a transformation between both spaces. The source may be an interval, a rectangle, or another measurable region. The target may hold length, area, density, probability, or an indicator weight. The calculator treats the target measure through a density when possible.

Why the Jacobian Matters

The Jacobian shows local stretching. In one dimension, it is the absolute derivative. For an affine rule T(x) = ax + b, the Jacobian is |a|. In two dimensions, it is the absolute determinant of the matrix. Large values expand measure. Small values shrink measure. A zero value means the map collapses volume.

Density Based View

Many advanced problems use a density instead of plain length or area. A Gaussian density gives more weight near its mean. A constant density applies the same weight everywhere. An indicator box counts only target points inside a chosen window. The calculator evaluates the target density after mapping each source point through the transformation.

Practical Use

Pullback measure is useful in probability, geometry, differential forms, integration, and change of variables. It helps explain how mass, probability, or area changes when coordinates are transformed. Students can test simple affine maps. Researchers can inspect numerical behavior for weighted regions. The graph makes density changes easier to see.

Numerical Accuracy

Exact formulas are used for affine Lebesgue and constant density cases. Gaussian and indicator choices may need numerical integration. Increase samples for smoother one dimensional estimates. Increase grid points for better two dimensional estimates. Very high settings can slow the page. Always compare results with known simple examples when precision matters.

FAQs

1. What is a pullback measure?

It is a measure moved from a target space back to a source space through a transformation. The calculator uses density and Jacobian scaling to estimate it.

2. What does the Jacobian do?

The Jacobian measures local stretching. In one dimension it is an absolute derivative. In two dimensions it is the absolute determinant.

3. Can this calculator handle Gaussian density?

Yes. It supports one dimensional Gaussian density and independent two dimensional Gaussian density. The result is computed numerically when needed.

4. What is the target window option?

It uses an indicator density. Points mapped inside the target interval or box get weight one. Other points get weight zero.

5. Why is the result zero for some maps?

A zero scale or zero determinant collapses the source volume. In this density formula, the pullback volume contribution becomes zero.

6. Which cases are exact?

Affine transformations with Lebesgue measure or constant density use exact formulas. Gaussian and indicator models usually use numerical integration.

7. How can I improve numerical accuracy?

Increase samples for one dimensional calculations. Increase grid points for two dimensional calculations. Avoid extreme values unless they are required.

8. Can I export the result?

Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple report of the calculated outputs.