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⨼

interior product

Interior Product

⨼ denotes contraction: it inserts a vector into a differential form, lowering the form's degree by one. In differential geometry, this operation is also called the interior product; it is different from an inner product of vectors.

Text symbol
Also called
contraction · insertion operator · interior multiplication
Search it as
iprod · intprod · contraction symbol · interior product notation · ⨼ vs ⨽
Shows up in
Anything that takes text · its shape is whatever your font draws

What ⨼ means

It inserts a vector into a differential form.

This operation is called contraction or the interior product. It puts a vector field into the form's first slot, leaving a form with one fewer slot. Interior product notation varies: the same operation is commonly written ι_X, where X identifies the vector field; ⨼ is another notation for it.

The history of the interior product belongs to a broader mathematical development: Hermann Grassmann's exterior algebra appeared in 1844, and Élie Cartan's formal work on differential forms began with a publication in 1899.

A specialist operator used in differential geometry, exterior algebra, and mathematical physics.

Don't confuse it with

1
⨽
Righthand Interior Product
Its hook faces the other way: ⨽ is the right-hand interior product. The direction determines which slot the vector occupies.

Questions people ask

2

Is ⨼ an inner product?

No. It contracts a vector with a differential form; an inner product pairs vectors and usually depends on a metric.

Why does ⨼ show as a box?

It can. A font that lacks this glyph may show a box even when ⨼ is present in the text; a font with mathematical operator coverage can display it.

How to type ⨼

Windows (Word)
Alt+XType 2A3C, then Alt+X to get ⨼.
Windows
Win+Rrun charmap for the full table, or use the emoji picker's Ω symbols tab (Win + .).
macOS
Control+Command+SpaceOpen the character viewer and search “interior product”.
Linux
Ctrl+Shift+UType 2a3c, then Enter or Space.
HTML
⨼
CSS
\2A3C
How to type ⨼
Some Unicode-aware LaTeX input tools insert ⨼ when you enter their configured command and press Space. The command varies by tool.
Codes and encoding1 code points
HTML decimal
⨼
HTML hex
⨼
CSS content
content: "\2A3C";
Each escape is closed by the next backslash; a trailing space is needed before a literal hex digit.
URL encoded
%E2%A8%BC
1
On screen
1
Code points
1
JS .length
3
UTF-8 bytes

A visible symbol can contain several code points. JS .length counts UTF-16 units; UTF-8 counts bytes.

Families this belongs to

1

Maths & numbers1,835 symbols

1️⃣📏📐✖️➕➖➗🟰♾️#️⃣*️⃣0️⃣