Differential Forms
Differential forms will allow us to express the fundamental theorem of calculus,
Green’s theorem, Stokes’ theorem, and the divergence theorem all as the same
theorem.
We have already encountered differential 1-forms (which were called
differentials in the first 3 semesters of calculus). For example, if 𝑧 = 𝑓(𝑥, 𝑦) we
have:
𝜕𝑓 𝜕𝑓
𝑑𝑧 = 𝑑𝑥 + 𝑑𝑦.
𝜕𝑥 𝜕𝑦
So if 𝑧 = 𝑥 2 + 𝑥𝑠𝑖𝑛𝑦,
𝑑𝑧 = (2𝑥 + 𝑠𝑖𝑛𝑦)𝑑𝑥 + (𝑥𝑐𝑜𝑠𝑦)𝑑𝑦.
The expression (2𝑥 + 𝑠𝑖𝑛𝑦)𝑑𝑥 + (𝑥𝑐𝑜𝑠𝑦)𝑑𝑦 is called a differential 1-form (or just a
1-form for short).
Real valued functions on an open set in ℝ3 (or ℝ𝑛 ) are called 0-forms. Thus
when we take the differential of a function (a 0-form) we get a 1-form. In fact, we
will see that we can define the operation of taking a differential of an 𝑛-form to get
an (𝑛 + 1)-form. We will then see that the fundamental theorem of calculus,
Green’s theorem, Stokes’ theorem, and the divergence theorem can all be written
as:
∫𝜕𝑀 𝜔 = ∫𝑀 𝑑𝜔,
where 𝜔 is a differential 𝑛-form and 𝑑𝜔 (the differential of 𝜔) is an 𝑛 + 1 form.
For the purposes of this section we will assume that all functions have as many
derivatives as we need.
, 2
0-Forms
Let 𝐾 be an open set in ℝ3 . A zero form on 𝐾 is a real valued function
𝑓: 𝐾 → ℝ. Given two 0-forms 𝑓1 and 𝑓2 on 𝐾, we can add them or multiply them.
Ex. Let 𝑓1 (𝑥, 𝑦, 𝑧) = 𝑥𝑒 𝑦𝑧 + 2𝑥𝑦, 𝑓2 (𝑥, 𝑦, 𝑧) = 𝑥𝑦. Then we have:
𝑓1 (𝑥, 𝑦, 𝑧) + 𝑓2 (𝑥, 𝑦, 𝑧) = 𝑥𝑒 𝑦𝑧 + 2𝑥𝑦 + 𝑥𝑦 = 𝑥𝑒 𝑦𝑧 + 3𝑥𝑦
[𝑓1 (𝑥, 𝑦, 𝑧)][𝑓2 (𝑥, 𝑦, 𝑧)] =( 𝑥𝑒 𝑦𝑧 + 2𝑥𝑦)(𝑥𝑦) = 𝑥 2 𝑦𝑒 𝑦𝑧 + 2𝑥 2 𝑦 2
1-Forms
A 1-form on 𝐾 ⊆ ℝ3 is of the form :
𝜔 = 𝑃(𝑥, 𝑦, 𝑧)𝑑𝑥 + 𝑄 (𝑥, 𝑦, 𝑧)𝑑𝑦 + 𝑅(𝑥, 𝑦, 𝑧)𝑑𝑧, or
𝜔 = 𝑃𝑑𝑥 + 𝑄𝑑𝑦 + 𝑅𝑑𝑧.
We line integrate 1-forms over a curve.
If we write: 𝜔 = 𝑄 (𝑥, 𝑦, 𝑧)𝑑𝑦, this is just a 1-form where
𝑃(𝑥, 𝑦, 𝑧) = 𝑅 (𝑥, 𝑦, 𝑧) = 0.
It also doesn’t matter which order we write the terms in:
𝜔 = 𝑃𝑑𝑥 + 𝑄𝑑𝑦 + 𝑅𝑑𝑧 = 𝑄𝑑𝑦 + 𝑅𝑑𝑧 + 𝑃𝑑𝑥.
However, the standard form is:
𝜔 = 𝑃𝑑𝑥 + 𝑄𝑑𝑦 + 𝑅𝑑𝑧.
Differential forms will allow us to express the fundamental theorem of calculus,
Green’s theorem, Stokes’ theorem, and the divergence theorem all as the same
theorem.
We have already encountered differential 1-forms (which were called
differentials in the first 3 semesters of calculus). For example, if 𝑧 = 𝑓(𝑥, 𝑦) we
have:
𝜕𝑓 𝜕𝑓
𝑑𝑧 = 𝑑𝑥 + 𝑑𝑦.
𝜕𝑥 𝜕𝑦
So if 𝑧 = 𝑥 2 + 𝑥𝑠𝑖𝑛𝑦,
𝑑𝑧 = (2𝑥 + 𝑠𝑖𝑛𝑦)𝑑𝑥 + (𝑥𝑐𝑜𝑠𝑦)𝑑𝑦.
The expression (2𝑥 + 𝑠𝑖𝑛𝑦)𝑑𝑥 + (𝑥𝑐𝑜𝑠𝑦)𝑑𝑦 is called a differential 1-form (or just a
1-form for short).
Real valued functions on an open set in ℝ3 (or ℝ𝑛 ) are called 0-forms. Thus
when we take the differential of a function (a 0-form) we get a 1-form. In fact, we
will see that we can define the operation of taking a differential of an 𝑛-form to get
an (𝑛 + 1)-form. We will then see that the fundamental theorem of calculus,
Green’s theorem, Stokes’ theorem, and the divergence theorem can all be written
as:
∫𝜕𝑀 𝜔 = ∫𝑀 𝑑𝜔,
where 𝜔 is a differential 𝑛-form and 𝑑𝜔 (the differential of 𝜔) is an 𝑛 + 1 form.
For the purposes of this section we will assume that all functions have as many
derivatives as we need.
, 2
0-Forms
Let 𝐾 be an open set in ℝ3 . A zero form on 𝐾 is a real valued function
𝑓: 𝐾 → ℝ. Given two 0-forms 𝑓1 and 𝑓2 on 𝐾, we can add them or multiply them.
Ex. Let 𝑓1 (𝑥, 𝑦, 𝑧) = 𝑥𝑒 𝑦𝑧 + 2𝑥𝑦, 𝑓2 (𝑥, 𝑦, 𝑧) = 𝑥𝑦. Then we have:
𝑓1 (𝑥, 𝑦, 𝑧) + 𝑓2 (𝑥, 𝑦, 𝑧) = 𝑥𝑒 𝑦𝑧 + 2𝑥𝑦 + 𝑥𝑦 = 𝑥𝑒 𝑦𝑧 + 3𝑥𝑦
[𝑓1 (𝑥, 𝑦, 𝑧)][𝑓2 (𝑥, 𝑦, 𝑧)] =( 𝑥𝑒 𝑦𝑧 + 2𝑥𝑦)(𝑥𝑦) = 𝑥 2 𝑦𝑒 𝑦𝑧 + 2𝑥 2 𝑦 2
1-Forms
A 1-form on 𝐾 ⊆ ℝ3 is of the form :
𝜔 = 𝑃(𝑥, 𝑦, 𝑧)𝑑𝑥 + 𝑄 (𝑥, 𝑦, 𝑧)𝑑𝑦 + 𝑅(𝑥, 𝑦, 𝑧)𝑑𝑧, or
𝜔 = 𝑃𝑑𝑥 + 𝑄𝑑𝑦 + 𝑅𝑑𝑧.
We line integrate 1-forms over a curve.
If we write: 𝜔 = 𝑄 (𝑥, 𝑦, 𝑧)𝑑𝑦, this is just a 1-form where
𝑃(𝑥, 𝑦, 𝑧) = 𝑅 (𝑥, 𝑦, 𝑧) = 0.
It also doesn’t matter which order we write the terms in:
𝜔 = 𝑃𝑑𝑥 + 𝑄𝑑𝑦 + 𝑅𝑑𝑧 = 𝑄𝑑𝑦 + 𝑅𝑑𝑧 + 𝑃𝑑𝑥.
However, the standard form is:
𝜔 = 𝑃𝑑𝑥 + 𝑄𝑑𝑦 + 𝑅𝑑𝑧.
