Einstein Summation Calculator

Calculator

Choose an operation, enter matching vectors or matrices, and submit the form.

Operation

Index base

Decimal places

Result scale factor

Custom notation label

Expanded terms

Vector a, x, or v

Vector b or w

Matrix A or metric g

Matrix B

Example Data Table

Use case	Notation	Input data	Expected idea
Dot product	a_i b_i	a = [2, -1, 4], b = [5, 3, 2]	Sum matching vector entries.
Trace	A_i^i	A = [[1, 2], [3, 4]]	Add diagonal entries.
Matrix product	A_ij B_jk	A is 2 × 3, B is 3 × 2	Contract the shared j index.
Metric contraction	g_ij v^i w^j	Square metric and two equal vectors	Return one scalar.

Formula Used

Einstein summation treats a repeated index as a summed index. The calculator uses finite arrays, so each repeated index is expanded as a standard sum.

Vector dot product: s = a_i b_i = Σ a_i b_i.
Trace: t = A_i^i = Σ A_ii.
Matrix-vector contraction: y_i = A_ij x_j = Σ A_ij x_j.
Matrix product: C_ik = A_ij B_jk = Σ A_ij B_jk.
Metric contraction: s = g_ij v^i w^j = ΣΣ g_ij v_i w_j.
Double contraction: s = A_ij B_ij = ΣΣ A_ij B_ij.
Levi-Civita contraction: c_i = ε_ijk a_j b_k.

How to Use This Calculator

Select the contraction or tensor operation.
Choose whether displayed indices start at zero or one.
Enter vectors as comma separated values.
Enter matrix rows on separate lines, or separate rows with semicolons.
Set decimal places and an optional scale factor.
Submit the form to show the result above the inputs.
Use the CSV or PDF buttons to save the calculation.

Einstein Summation in Modern Mathematics

Einstein summation is a compact way to write repeated sums. It is common in tensor algebra, relativity, continuum mechanics, and differential geometry. A repeated index means that all allowed values are added. This removes long sigma symbols. It also shows the structure of a calculation more clearly.

Why the Convention Helps

The notation connects vectors, matrices, and higher tensors through shared indices. A dot product becomes a_i b_i. A trace becomes A_i^i. A matrix product becomes A_ij B_jk. Each repeated index is internal. Each unrepeated index remains in the final result. This makes formulas shorter. It also reduces errors in complex derivations.

Practical Calculator Uses

This calculator supports common contractions used in mathematics and physics. You can evaluate vector products, matrix traces, matrix-vector products, matrix multiplication, metric contractions, double contractions, and cross products. Each operation includes dimension checks. The tool also shows expanded steps. That helps learners see which index is summed.

Checking Tensor Work

Tensor notation is powerful, but mistakes are easy. A repeated index may appear too many times. A matrix may have the wrong shape. A metric tensor may not match the vector dimension. This calculator flags those issues before producing a result. It keeps the calculation transparent and repeatable.

Learning With Examples

Example data helps you compare notation with numeric output. Try a dot product first. Then test a trace or matrix product. Notice how the summed index disappears from the result. For metric contraction, the scalar depends on the metric entries. For the Levi-Civita option, only three-dimensional vectors are valid.

Exporting Results

Download options make the calculator useful for notes and reports. The CSV file stores numeric results and steps. The PDF file gives a readable summary. Both exports keep the chosen notation, input data, and final answer together. This makes review easier during assignments, research checks, or classroom demonstrations.

Reliable Study Workflow

Use the calculator as a verification aid. Start with clean input. Check dimensions. Read the expanded expression. Compare the answer with your manual work. It encourages careful naming of free indices. Consistent names make tensor equations easier to audit, explain, and reuse in later calculations. Repeat with different tensors to strengthen your understanding of index notation.

FAQs

What is Einstein summation?

It is a notation rule. A repeated index means that values are summed over that index. It shortens vector, matrix, and tensor formulas.

Which inputs are needed for a dot product?

Enter vector a and vector b. Both vectors must have the same number of components, because matching entries are multiplied and added.

How should I enter a matrix?

Write one row per line. You may also separate rows with semicolons. Use commas or spaces between values inside each row.

What does a free index mean?

A free index is not summed. It remains in the output. For example, y_i has one free index, so the result is a vector.

Why did the calculator reject my matrices?

The dimensions may not match the chosen contraction. For matrix multiplication, columns of A must equal rows of B. For trace, A must be square.

Can this handle metric tensors?

Yes. Select metric contraction. Enter the metric as matrix A. Then enter vectors v and w in the two vector boxes.

What does the scale factor do?

It multiplies the raw result after the contraction is computed. Use it for constants, coefficients, unit conversions, or formula adjustments.

Are the downloads calculated again?

Yes. The export buttons resubmit the same inputs. The server rebuilds the same result and returns either a CSV file or a compact PDF.