Addition and subtraction

If ${\displaystyle \mathbf {a} \,}$ and ${\displaystyle \mathbf {b} \,}$ are vectors, then the sum ${\displaystyle \mathbf {c} =\mathbf {a} +\mathbf {b} \,}$ is also a vector (see Figure 2(a)).

The two vectors can also be subtracted from one another to give another vector ${\displaystyle \mathbf {d} =\mathbf {a} -\mathbf {b} \,}$.

Multiplication by a scalar

Multiplication of a vector ${\displaystyle \mathbf {b} \,}$ by a scalar ${\displaystyle \lambda \,}$ has the effect of stretching or shrinking the vector (see Figure 2(b)).

You can form a unit vector ${\displaystyle {\hat {\mathbf {b} }}\,}$ that is parallel to ${\displaystyle \mathbf {b} \,}$ by dividing by the length of the vector ${\displaystyle |\mathbf {b} |\,}$. Thus,

${\displaystyle {\hat {\mathbf {b} }}={\cfrac {\mathbf {b} }{|\mathbf {b} |}}~.}$

Scalar product of two vectors

The scalar product or inner product or dot product of two vectors is defined as

${\displaystyle \mathbf {a} \bullet \mathbf {b} =|\mathbf {a} ||\mathbf {b} |\cos(\theta )}$

where ${\displaystyle \theta \,}$ is the angle between the two vectors (see Figure 2(b)).

If ${\displaystyle \mathbf {a} \,}$ and ${\displaystyle \mathbf {b} \,}$ are perpendicular to each other, ${\displaystyle \theta =\pi /2\,}$ and ${\displaystyle \cos(\theta )=0\,}$. Therefore, ${\displaystyle {\mathbf {a} }\bullet {\mathbf {b} }=0}$.

The dot product therefore has the geometric interpretation as the length of the projection of ${\displaystyle \mathbf {a} \,}$ onto the unit vector ${\displaystyle {\hat {\mathbf {b} }}\,}$ when the two vectors are placed so that they start from the same point.

The scalar product leads to a scalar quantity and can also be written in component form (with respect to a given basis) as

${\displaystyle {\mathbf {a} }\bullet {\mathbf {b} }=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}=\sum _{i=1..3}a_{i}b_{i}~.}$

If the vector is ${\displaystyle n}$ dimensional, the dot product is written as

${\displaystyle {\mathbf {a} }\bullet {\mathbf {b} }=\sum _{i=1..n}a_{i}b_{i}~.}$

Using the Einstein summation convention, we can also write the scalar product as

${\displaystyle {\mathbf {a} }\bullet {\mathbf {b} }=a_{i}b_{i}~.}$

Also notice that the following also hold for the scalar product

1. ${\displaystyle {\mathbf {a} }\bullet {\mathbf {b} }={\mathbf {b} }\bullet {\mathbf {a} }}$ (commutative law).
2. ${\displaystyle {\mathbf {a} }\bullet {(\mathbf {b} +\mathbf {c} )}={\mathbf {a} }\bullet {\mathbf {b} }+{\mathbf {a} }\bullet {\mathbf {c} }}$ (distributive law).

Vector product of two vectors

The vector product (or cross product) of two vectors ${\displaystyle \mathbf {a} \,}$ and ${\displaystyle \mathbf {b} \,}$ is another vector ${\displaystyle \mathbf {c} \,}$ defined as

${\displaystyle \mathbf {c} ={\mathbf {a} }\times {\mathbf {b} }=|\mathbf {a} ||\mathbf {b} |\sin(\theta ){\hat {\mathbf {c} }}}$

where ${\displaystyle \theta \,}$ is the angle between ${\displaystyle \mathbf {a} \,}$ and ${\displaystyle \mathbf {b} \,}$, and ${\displaystyle {\hat {\mathbf {c} }}\,}$ is a unit vector perpendicular to the plane containing ${\displaystyle \mathbf {a} \,}$ and ${\displaystyle \mathbf {b} \,}$ in the right-handed sense (see Figure 3 for a geometric interpretation)

Figure 3: Vector product of two vectors.

In terms of the orthonormal basis ${\displaystyle (\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3})\,}$, the cross product can be written in the form of a determinant

${\displaystyle {\mathbf {a} }\times {\mathbf {b} }={\begin{vmatrix}\mathbf {e} _{1}&\mathbf {e} _{2}&\mathbf {e} _{3}\\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\end{vmatrix}}~.}$

In index notation, the cross product can be written as

${\displaystyle {\mathbf {a} }\times {\mathbf {b} }\equiv \varepsilon _{ijk}\mathbf {e} _{i}a_{j}b_{k}~.}$

where ${\displaystyle \varepsilon _{ijk}}$ is the Levi-Civita symbol (also called the permutation symbol, alternating tensor).

Derivative of a vector valued function

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

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

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

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

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

If ${\displaystyle \mathbf {a} (x)\,}$ and ${\displaystyle \mathbf {b} (x)\,}$ are two vector functions, then from the chain rule we get

${\displaystyle {\begin{aligned}{\cfrac {d({\mathbf {a} }\bullet {\mathbf {b} })}{dx}}&={\mathbf {a} }\bullet {\cfrac {d\mathbf {b} }{dx}}+{\cfrac {d\mathbf {a} }{dx}}\bullet {\mathbf {b} }\\{\cfrac {d({\mathbf {a} }\times {\mathbf {b} })}{dx}}&={\mathbf {a} }\times {\cfrac {d\mathbf {b} }{dx}}+{\cfrac {d\mathbf {a} }{dx}}\times {\mathbf {b} }\\{\cfrac {d[{\mathbf {a} }\bullet {({\mathbf {b} }\times {\mathbf {c} })}]}{dx}}&={\cfrac {d\mathbf {a} }{dx}}\bullet {({\mathbf {b} }\times {\mathbf {c} })}+{\mathbf {a} }\bullet {\left({\cfrac {d\mathbf {b} }{dx}}\times {\mathbf {c} }\right)}+{\mathbf {a} }\bullet {\left({\mathbf {b} }\times {\cfrac {d\mathbf {c} }{dx}}\right)}\end{aligned}}}$

Scalar and vector fields

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

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

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 (${\displaystyle \mathbf {a} \,}$) that assigns a vector to each point in space, then

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

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

Gradient of a scalar field

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

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

In index notation,

${\displaystyle {\boldsymbol {\nabla }}{\varphi }\equiv \varphi _{,i}~\mathbf {e} _{i}~.}$

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

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

Divergence of a vector field

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

${\displaystyle {\boldsymbol {\nabla }}\bullet {\mathbf {u} }={\frac {\partial u_{1}}{\partial x_{1}}}+{\frac {\partial u_{2}}{\partial x_{2}}}+{\frac {\partial u_{3}}{\partial x_{3}}}~.}$

In index notation,

${\displaystyle {\boldsymbol {\nabla }}\bullet {\mathbf {u} }\equiv u_{i,i}~.}$

If ${\displaystyle {\boldsymbol {\nabla }}\bullet {\mathbf {u} }=0}$, then ${\displaystyle \mathbf {u} \,}$ 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

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

${\displaystyle {\boldsymbol {\nabla }}\times {\mathbf {u} }=\det {\begin{vmatrix}\mathbf {e} _{1}&\mathbf {e} _{2}&\mathbf {e} _{3}\\{\frac {\partial }{\partial x_{1}}}&{\frac {\partial }{\partial x_{2}}}&{\frac {\partial }{\partial x_{3}}}\\u_{1}&u_{2}&u_{3}\\\end{vmatrix}}}$

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

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

${\displaystyle \nabla ^{2}{\varphi }:={\boldsymbol {\nabla }}\bullet {{\boldsymbol {\nabla }}{\varphi }}={\frac {\partial ^{2}\varphi }{\partial x_{1}^{2}}}+{\frac {\partial ^{2}\varphi }{\partial x_{2}^{2}}}+{\frac {\partial ^{2}\varphi }{\partial x_{3}^{2}}}~.}$

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

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

Green-Gauss divergence theorem

Let ${\displaystyle \mathbf {u} (\mathbf {x} )\,}$ be a continuous and differentiable vector field on a body ${\displaystyle \Omega \,}$ with boundary ${\displaystyle \Gamma \,}$. The divergence theorem states that

${\displaystyle {\int _{\Omega }{\boldsymbol {\nabla }}\bullet {\mathbf {u} }~dV=\int _{\Gamma }{\mathbf {n} }\bullet {\mathbf {u} }~dA}}$

where ${\displaystyle \mathbf {n} \,}$ is the outward unit normal to the surface (see Figure 5).

In index notation,

${\displaystyle \int _{\Omega }u_{i,i}~dV=\int _{\Gamma }n_{i}u_{i}~dA}$

Figure 5: Volume for application of the divergence theorem.