Vector calculus

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Material from Vectors was moved[1] here.

Here we extend the concept of vector to that of the vector field. A familiar example of a vector field is wind velocity: It has direction and magnitude, which makes it a vector. But it also depends on position (and ultimately on time). Wind velocity is a function of (x,y,z) at any given time, equivalently we can say that wind velocity is a time-dependent field:  V -> w i n d = V -> ( r -> , t ) {\displaystyle {\vec {V}}_{wind}={\vec {V}}({\vec {r}},t)} .

### Derivative of a vector valued function

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Let  a ( x ) {\displaystyle \mathbf {a} (x)\,} be a vector function that can be represented as

a ( x ) = a 1 ( x ) e 1 + a 2 ( x ) e 2 + a 3 ( x ) e 3 {\displaystyle \mathbf {a} (x)=a_{1}(x)\mathbf {e} _{1}+a_{2}(x)\mathbf {e} _{2}+a_{3}(x)\mathbf {e} _{3}\,}

where  x {\displaystyle x\,} is a scalar.

Then the derivative of  a ( x ) {\displaystyle \mathbf {a} (x)\,} with respect to  x {\displaystyle x\,} is

d a ( x ) d x = lim Δ x -> 0 a ( x + Δ x ) − a ( x ) Δ x = d a 1 ( x ) d x e 1 + d a 2 ( x ) d x e 2 + d a 3 ( x ) d x e 3 . {\displaystyle {\cfrac {d\mathbf {a} (x)}{dx}}=\lim _{\Delta x\rightarrow 0}{\cfrac {\mathbf {a} (x+\Delta x)-\mathbf {a} (x)}{\Delta x}}={\cfrac {da_{1}(x)}{dx}}\mathbf {e} _{1}+{\cfrac {da_{2}(x)}{dx}}\mathbf {e} _{2}+{\cfrac {da_{3}(x)}{dx}}\mathbf {e} _{3}~.}

Note: In the above equation, the unit vectors  e i {\displaystyle \mathbf {e} _{i}} (i=1,2,3) are assumed constant.
If  a ( x ) {\displaystyle \mathbf {a} (x)\,} and  b ( x ) {\displaystyle \mathbf {b} (x)\,} are two vector functions, then from the chain rule we get

d ( a ⋅ b ) x = a ⋅ d b d x + d a d x ⋅ b d ( a × b ) d x = a × d b d x + d a d x × b d [ a ⋅ ( a × b ) ] d t = d a d t ⋅ ( b × c ) + a ⋅ ( d b d t × c ) + a ⋅ ( b × d c d t ) {\displaystyle {\begin{aligned}{\cfrac {d({\mathbf {a} }\cdot {\mathbf {b} })}{x}}&={\mathbf {a} }\cdot {\cfrac {d\mathbf {b} }{dx}}+{\cfrac {d\mathbf {a} }{dx}}\cdot {\mathbf {b} }\\\\{\cfrac {d({\mathbf {a} }\times {\mathbf {b} })}{dx}}&={\mathbf {a} }\times {\cfrac {d\mathbf {b} }{dx}}+{\cfrac {d\mathbf {a} }{dx}}\times {\mathbf {b} }\\\\{\cfrac {d[{\mathbf {a} }\cdot {({\mathbf {a} }\times {\mathbf {b} })}]}{dt}}&={\cfrac {d\mathbf {a} }{dt}}\cdot {({\mathbf {b} }\times {\mathbf {c} })}+{\mathbf {a} }\cdot {\left({\cfrac {d\mathbf {b} }{dt}}\times {\mathbf {c} }\right)}+{\mathbf {a} }\cdot {\left({\mathbf {b} }\times {\cfrac {d\mathbf {c} }{dt}}\right)}\end{aligned}}}

### Scalar and vector fields

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Let  x {\displaystyle \mathbf {x} \,} be the position vector of any point in space. Suppose that there is a scalar function ( g {\displaystyle g\,} ) that assigns a value to each point in space. Then

g = g ( x ) {\displaystyle g=g(\mathbf {x} )\,}

represents a scalar field. An example of a scalar field is the temperature. See Figure4(a).

Figure 4: Scalar and vector fields.

If there is a vector function ( a {\displaystyle \mathbf {a} \,} ) that assigns a vector to each point in space, then

a = a ( x ) {\displaystyle \mathbf {a} =\mathbf {a} (\mathbf {x} )\,}

represents a vector field. An example is the displacement field. See Figure 4(b).

### Gradient of a scalar field

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Let  φ ( x ) {\displaystyle \varphi (\mathbf {x} )\,} be a scalar function. Assume that the partial derivatives of the function are continuous in some region of space. If the point  x {\displaystyle \mathbf {x} \,} has coordinates ( x 1 , x 2 , x 3 {\displaystyle x_{1},x_{2},x_{3}\,} ) with respect to the basis ( e 1 , e 2 , e 3 {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}\,} ), the gradient of  φ {\displaystyle \varphi \,} is defined as

∇ φ = ∂ φ ∂ x 1 e 1 + ∂ φ ∂ x 2 e 2 + ∂ φ ∂ x 3 e 3 . {\displaystyle {\boldsymbol {\nabla }}{\varphi }={\frac {\partial \varphi }{\partial x_{1}}}~\mathbf {e} _{1}+{\frac {\partial \varphi }{\partial x_{2}}}~\mathbf {e} _{2}+{\frac {\partial \varphi }{\partial x_{3}}}~\mathbf {e} _{3}~.}

In index notation,

∇ φ ≡ φ , i e i . {\displaystyle {\boldsymbol {\nabla }}{\varphi }\equiv \varphi _{,i}~\mathbf {e} _{i}~.}

The gradient is obviously a vector and has a direction. We can think of the gradient at a point being the vector perpendicular to the level contour at that point.

It is often useful to think of the symbol  ∇ {\displaystyle {\boldsymbol {\nabla }}{}} as an operator of the form

∇ = ∂ ∂ x 1 e 1 + ∂ ∂ x 2 e 2 + ∂ ∂ x 3 e 3 . {\displaystyle {\boldsymbol {\nabla }}{}={\frac {\partial }{\partial x_{1}}}~\mathbf {e} _{1}+{\frac {\partial }{\partial x_{2}}}~\mathbf {e} _{2}+{\frac {\partial }{\partial x_{3}}}~\mathbf {e} _{3}~.}

### Divergence of a vector field

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If we form a scalar product of a vector field  u ( x ) {\displaystyle \mathbf {u} (\mathbf {x} )\,} with the  ∇ {\displaystyle {\boldsymbol {\nabla }}{}} operator, we get a scalar quantity called the divergence of the vector field. Thus,

∇ ⋅ u = ∂ u 1 ∂ x 1 + ∂ u 2 ∂ x 2 + ∂ u 3 ∂ x 3 . {\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {u} ={\frac {\partial u_{1}}{\partial x_{1}}}+{\frac {\partial u_{2}}{\partial x_{2}}}+{\frac {\partial u_{3}}{\partial x_{3}}}~.}

In index notation,

∇ u ≡ u i , i . {\displaystyle {\boldsymbol {\nabla }}\mathbf {u} \equiv u_{i,i}~.}

If  ∇ ⋅ u = 0 {\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {u} =0} , then  u {\displaystyle \mathbf {u} \,} is called a divergence-free field.

The physical significance of the divergence of a vector field is the rate at which some density exits a given region of space. In the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or out of the region.

### Curl of a vector field

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The curl of a vector field  u ( x ) {\displaystyle \mathbf {u} (\mathbf {x} )\,} is a vector whose expression can be obtained with

∇ × u = | e 1 e 2 e 3 ∂ ∂ x 1 ∂ ∂ x 2 ∂ ∂ x 3 u 1 u 2 u 3 | {\displaystyle {\boldsymbol {\nabla }}\times {\mathbf {u} }={\begin{vmatrix}\mathbf {e} _{1}&\mathbf {e} _{2}&\mathbf {e} _{3}\\\\{\frac {\partial }{\partial x_{1}}}&{\frac {\partial }{\partial x_{2}}}&{\frac {\partial }{\partial x_{3}}}\\\u_{1}&u_{2}&u_{3}\\\\\end{vmatrix}}}

The physical significance of the curl of a vector field is the amount of rotation or angular momentum of the contents of a region of space.

### Laplacian of a scalar or vector field

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The Laplacian of a scalar field  φ ( x ) {\displaystyle \varphi (\mathbf {x} )\,} is a scalar defined as

∇ 2 φ := ∇ ( ∇ φ ) = ∂ 2 φ ∂ x 1 + ∂ 2 φ ∂ x 2 + ∂ 2 φ ∂ x 3 . {\displaystyle \nabla ^{2}{\varphi }:={\boldsymbol {\nabla }}({\boldsymbol {\nabla }}{\varphi })={\frac {\partial ^{2}\varphi }{\partial x_{1}}}+{\frac {\partial ^{2}\varphi }{\partial x_{2}}}+{\frac {\partial ^{2}\varphi }{\partial x_{3}}}~.}

The Laplacian of a vector field  u ( x ) {\displaystyle \mathbf {u} (\mathbf {x} )\,} is a vector defined as

∇ 2 u := ( ∇ 2 u 1 ) e 1 + ( ∇ 2 u 2 ) e 2 + ( ∇ 2 u 3 ) e 3 . {\displaystyle \nabla ^{2}{\mathbf {u} }:=(\nabla ^{2}{u_{1}})\mathbf {e} _{1}+(\nabla ^{2}{u_{2}})\mathbf {e} _{2}+(\nabla ^{2}{u_{3}})\mathbf {e} _{3}~.}

### Identities in vector calculus

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Some frequently used identities from vector calculus are listed below.

1. ∇ ( a + b ) = ∇ ⋅ a + ∇ ⋅ b {\displaystyle {\boldsymbol {\nabla }}(\mathbf {a} +\mathbf {b} )={\boldsymbol {\nabla }}\cdot {\mathbf {a} }+{\boldsymbol {\nabla }}\cdot {\mathbf {b} }}
2. ∇ × ( a + b ) = ∇ × a + ∇ × b {\displaystyle {\boldsymbol {\nabla }}\times {(\mathbf {a} +\mathbf {b} )}={\boldsymbol {\nabla }}\times {\mathbf {a} }+{\boldsymbol {\nabla }}\times {\mathbf {b} }}
3. ∇ ( φ a ) = ⋅ ( ∇ φ ) a + φ ( ∇ ⋅ a ) {\displaystyle {\boldsymbol {\nabla }}(\varphi \mathbf {a} )=\cdot {({\boldsymbol {\nabla }}{\varphi })}{\mathbf {a} }+\varphi ({\boldsymbol {\nabla }}\cdot {\mathbf {a} })}
4. ∇ × ( φ a ) = ( ∇ φ ) × a + φ ( ∇ × a ) {\displaystyle {\boldsymbol {\nabla }}\times {(\varphi \mathbf {a} )}={({\boldsymbol {\nabla }}{\varphi })}\times {\mathbf {a} }+\varphi ({\boldsymbol {\nabla }}\times {\mathbf {a} })}
5. ∇ ( a × b ) = b ⋅ ( ∇ × a ) − a ⋅ ( ∇ × b ) {\displaystyle {\boldsymbol {\nabla }}({\mathbf {a} }\times {\mathbf {b} })={\mathbf {b} }\cdot {({\boldsymbol {\nabla }}\times {\mathbf {a} })}-{\mathbf {a} }\cdot {({\boldsymbol {\nabla }}\times {\mathbf {b} })}}

## Fundamental theorems of vector calculus

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One version of the fundamental theorem of one-dimensional calculus is

∫ a b f ′ ( x ) d x = f ( b ) − f ( a ) {\displaystyle \int _{a}^{b}f\,'(x)dx=f(b)-f(a)}

This is a theorem about a function,  f ( x ) {\displaystyle f(x)} , its first derivative, and a line segment. Two notations used to denote this line segment are [a,b] and the inequality, a<x<b.  In the field of topology,  ∂ {\displaystyle \partial } denotes boundary. If we let the symbol  L {\displaystyle {\mathcal {L}}} denote the infinite number of points in the line segment [a,b], then the symbol  ∂ L {\displaystyle \partial {\mathcal {L}}} denotes the two endpoints (at  x = a  and  x = b ) of the line segment  L {\displaystyle {\mathcal {L}}} . These endpoints form the boundary of  L {\displaystyle {\mathcal {L}}} .

### Gradient theorem

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The gradient theorem is a direct generalization of the fundamental theorem of calculus:

∫ ℓ [ p -> -> q -> ] ⊂ R n ∇ -> f ⋅ d ℓ -> = f ( q -> ) − f ( p -> ) {\displaystyle \int _{\ell [{\vec {p}}\to {\vec {q}}]\subset \mathbb {R} ^{n}}{\vec {\nabla }}f\cdot d{\vec {\ell }}=f\left({\vec {q}}\right)-f\left({\vec {p}}\right)}

The subscript,  ℓ [ p -> -> q -> ] ⊂ R n {\displaystyle \ell [{\vec {p}}\to {\vec {q}}]\subset \mathbb {R} ^{n}} informs this is an integral over the over a one-dimensinal curve (or 'path') line integral ℓ {\displaystyle \ell } from point  r -> = p -> {\displaystyle {\vec {r}}={\vec {p}}} to point  r -> = q -> {\displaystyle {\vec {r}}={\vec {q}}} . The function,  f = f ( r -> ) {\displaystyle f=f({\vec {r}})} is any scalar field that is differentiable. The expression  ⊂ R n {\displaystyle \subset \mathbb {R} ^{n}} informs us that  r -> {\displaystyle {\vec {r}}} can be a member of an n-dimensional space. (In other words the theorem is easily generalized to more than three dimensions.) A consequence of this theorem is that  ∫ ∇ -> f ⋅ d ℓ -> = 0 {\displaystyle \int {\vec {\nabla }}f\cdot d{\vec {\ell }}=0} for any "closed curve" The figure shows the closed curve A, as well as the "open curve", B. Two endpoints form the "boundary" of curve B.

### Stokes' theorem

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Stokes' theorem states:

∫ Σ ⊂ R 3 ∇ -> × F -> ⋅ d A -> = ∮ ∂ Σ F -> ⋅ d ℓ ^ {\displaystyle \int _{\Sigma \subset \mathbb {R} ^{3}}{\vec {\nabla }}\times {\vec {F}}\cdot {\vec {dA}}=\oint _{\partial \Sigma }{\vec {F}}\cdot d{\hat {\ell }}}

The integral subscript,  Σ ⊂ R 3 {\displaystyle \Sigma \subset \mathbb {R} ^{3}} informs us that this theorem is valid only in a three-dimensional vector space. The integral is over a two-dimensional surface,Σ ,with  d A -> = n ^ d A {\displaystyle {\vec {dA}}={\hat {n}}dA} , where  n ^ {\displaystyle {\hat {n}}} is normal to the surface. The integral over the surface, Σ, is nonzero only if its boundary, ∂Σ, exists. Surfaces with such boundaries are called open surfaces, and the boundary, ∂Σ, is a curve in 3-space that goes along the "edge" of the surface. This curve is integrated in the direction of positive orientation, meaning that  d ℓ -> {\displaystyle {\vec {d\ell }}} and the surface normal follow  n ^ d A {\displaystyle {\hat {n}}dA} follow the right-hand rule.

Footnote: According to Wikipedia[2], this form of the theorem was first discovered by Lord Kelvin, who communicated it to George Stokes in a letter dated July 2, 1850. Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name.

### Divergence theorem

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closed  | open:  Ω & ∂Ω

The divergence theorem states:

∫ Ω ∇ -> ⋅ F -> d V = ∮ ∂ Ω ⊂ R 3 F -> ⋅ d A -> {\displaystyle \int _{\Omega }{\vec {\nabla }}\cdot {\vec {F}}\;dV=\oint _{\partial \Omega \subset \mathbb {R} ^{3}}{\vec {F}}\cdot d{\vec {A}}} .

The integral subscript,  Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} , informs us that this theorem is valid in an (arbitrary) n-dimensional vector space. The n-dimensional volume is Ω, and ∂Ω is its boundary. If n =3 dimensions, ∂Ω is a surface. Since this surface encloses a volume, it has no boundary of its own, and is therefore called a closed surface. The figure shows six surfaces. The three on the left have no boundary and are therefore closed; the ones to the right have a boundary (shown in red) and are therefore open. Note that the closed surfaces to the left are themselves boundaries volumes which are defined as what is "inside" the surface.

Footnote: In index notation, the gradient theorem can be written as  ∫ Ω u i , i d V = ∫ ∂ Ω n i u i d A {\displaystyle \int _{\Omega }u_{i,i}~dV=\int _{\partial \Omega }n_{i}u_{i}~dA}

1. ↑ https://en.wikiversity.orghttps://en.wikiversity.org/w/index.php?title=Vectors&oldid=1194637
2. ↑ https://en.wikipedia.orghttps://en.wikiversity.org/w/index.php?title=Stokes'_theorem&oldid=608564969