█████████ █████

██████████ ███████

In ███████████, ████████ ████████ (████ ██████ █████████ ████████) █████████ █████ █████ in ███-███████████ █████ by █████ of ███ ███████████. ███ █████████████ ██████████ ████ are █████ which █████ ████ █████ at the ██████. They are ███████ ███████ █ and █. ████████ to █████ ████, the ████████ of ███ █████ in ███-███████████ █████ is █████ by an ███████ ████ of ████ ███████, ████ ██████ ██████ the ████████ of that █████ from the ██████ ████████ along the █████ ████, which is █████ to the ████████ of that █████ from the █████ ████.

███████ ██████ used ██████████ ██████ is the █████ ██████████ ██████, which █████████ a █████ in █████ of ███ ████████ from the ██████ and ███ █████ ████████ to a █████████ █████████ ███.

█████████ in █████-███████████ █████

█████████

In ███ ██████████, there are ██████████ ████ █████████: the ████████. The █████ ███ ███████ ████ are █████ below:

██████

The ███████████ in █ ██████████ is a ██████, █████████ ██████ a █-██████ (█^█) because it is a ███-███████████ ████████. In a █████████ █████, it has the ██████ ███ and the ████ of ███ ████████ is

${\displaystyle A=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$

█████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is the ██████.

█████ ██████

There are an ██████████ of █████ ██████ ██████ in ███ ██████████, ███████ █████████ the █████ ████████: the ███████, the ████████, and the █████████.

███ ███████, █████, and ██████

The ███ ███████ of ███ ███████ A = [A_█, A_█] and █ = [█_█, █_█] is ███████ as:

${\displaystyle \mathbf{A}\cdot \mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}=A_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}+A_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$

A ██████ can be ████████ as an █████. ███ █████████ is ███ ██████, and ███ █████████ is the █████████ the █████ ██████. The █████████ of a ██████ A is ███████ by ${\displaystyle \Vert \mathbf{A}\Vert }$. In this █████████, the ███ ███████ of ███ █████████ ███████ A and █ is ███████ by

${\displaystyle \mathbf{A}\cdot \mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}=\Vert \mathbf{A}\Vert \Vert \mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}\Vert \mathrm{<span\; word-length="3"\; class="word\; redacted">███</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},}$

█████ █ is the █████ between A and █.

The ███ ███████ of a ██████ A by ██████ is

${\displaystyle \mathbf{A}\cdot \mathbf{A}=\Vert \mathbf{A}{\Vert }^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},}$

which █████

${\displaystyle \Vert \mathbf{A}\Vert =\sqrt{\mathbf{A}\cdot \mathbf{A}},}$

the ███████ for the █████████ ██████ of the ██████.

████████

In a ███████████ ██████████ ██████, the ████████ is █████ by

${\displaystyle \mathrm{\nabla }\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\frac{\mathrm{\partial }\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{\partial }\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}+\frac{\mathrm{\partial }\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{\partial }\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}.}$

████ █████████ and ██████ █████████

For some ██████ █████ █ : █ ⊆ █^█ → █, the ████ ████████ along a █████████ ██████ █████ █ ⊂ █ is ███████ as

${\displaystyle \underset{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\int }\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}={\int }_a^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}))|{\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\prime }(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})|\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},}$

█████ █: [a, █] → █ is an █████████ █████████ ███████████████ of the █████ █ ████ that █(a) and █(█) ████ the █████████ of █ and .

For a ██████ █████ █ : █ ⊆ █^█ → █^█, the ████ ████████ along a █████████ ██████ █████ █ ⊂ █, in the █████████ of █, is ███████ as

${\displaystyle \underset{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\int }\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>})\cdot \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}={\int }_a^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}))\cdot {\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\prime }(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},}$

█████ · is the ███ ███████ and █: [a, █] → █ is a █████████ ███████████████ of the █████ █ ████ that █(a) and █(█) ████ the █████████ of █.

A ██████ ████████ ██████ to an ████████ within a ██████ █ in █^█ of a ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}),}$ and is ███████ ███████ as:

${\displaystyle \underset{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\iint }\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}.}$

███████████ ███████ of ████ █████████

The ███████████ ███████ of ████ █████████ ████ that a ████ ████████ through a ████████ █████ can be █████████ by ██████████ the ████████ ██████ █████ at the █████████ of the █████.

███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\subseteq {\mathbb{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\to \mathbb{<span\; word-length="1"\; class="word\; redacted">█</span>}}$. ████

${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\left(\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}\right)-\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\left(\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}\right)={\int }_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}[\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}]}\mathrm{\nabla }\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>})\cdot \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>},}$

with █, █ the █████████ of the █████ █.

█████'s ███████

███ █ be a ██████████ ████████, █████████ ██████, ██████ ██████ █████ in a █████, and ███ █ be the ██████ ███████ by █. If █ and █ are █████████ of (█, █) ███████ on an ████ ██████ ██████████ █ and have ██████████ ███████ ███████████ there, ████

${\displaystyle {\oint }_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}+\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})={\iint }_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\left(\frac{\mathrm{\partial }\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{\partial }\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}-\frac{\mathrm{\partial }\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{\partial }\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\right)\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$

█████ the ████ of ███████████ along █ is ████████████████.