Line Integrals

Line integrals are used to calculate the integral of a function along a path in space rather than over a volume as can be done with Density and Mass integrals

Definition

If $f$ is defined on a curve $C$ given parametrically by $r(t)=g(t)i+h(t)j+k(t)k,a\le t\le b$ then the line integral is given by

$$\int \_Cf(x,y,z)ds=\int \_a^{b}f(g(t),h(t),k(t))|v(t)|(dot)$$

• where $v(t)=r^{\prime }(t)=ds/dt$ (velocity vector is necessary because it gives the magnitude → distance of travel)
• The function $f$ inside the integrand is parametrized with $r(t)$ because it is based on position
• If $f$ happens to have the constant value 1, the integral will evaluate the length of the path

Conditions

• $r(t)$ is a smooth parametrization of C

Additivity

A line integral is the sum of the integrals of all the lines that make the entire path up

$$\int \_C=\int \_C\_1+...+\int \_C\_n$$

Line Integrals for 2D-Surfaces

When integrating a function $f(x,y)=z$ over a line segment $C$ the resulting Area will be the area enclosed by the xy-plane and the z-value of the function at that point

$$A=\int \_Cfds$$

Line Integrals in Vector Fields

If the parametrization of the path $C$ is continuous, at each point along the path $C$, the tangent vector $T=dr/ds=v/|v|$ is a unit vector tangent to the path and pointing to the forward direction of it.

Let $F(x,y,z)$ be a continuous Vector Field

The line integral of the vector field is the line integral of the scalar tangential component of $F$ along $C$. This tangential component is given by

$$F\cdot T=F\cdot \frac{dr}{ds}$$

Hence the line integral is

$$\int \_CF\cdot Tds=\int \_CF\cdot dr$$

Best practice is to always parametrize the vector field’s equation as well

$$\int \_CF\cdot dr=\int \_a^{b}F(r(t))\cdot \frac{dr}{dt}dt$$

One Directional Vector Field Integrals

When integrating with respect to one direction only (to determine a force or flow in it disregarding every other direction) the vector Field $F$ should contain only the components associated with the directions of integration such as $F=M(x,y,z)i$ (x-direction). For the parametrized curve $r(t)=g(t)i+h(t)j+...$ we have $x=g(t)$ and $dx=g^{\prime }(t)$ such that

$$F\cdot dr=M(x,y,z)i\cdot g^{\prime }(t)dt=M(x,y,z)dx$$

the line integral becomes

$$\int \_CF\cdot dr=\int \_CM(x,y,z)dx$$

Work along Line Integrals

If we consider the vector field to be a field of force, then the work being done across the path C in it is given by the line integral

$$\int \_CF\cdot Tds$$

which now defines the total work done (for example the force field moving an object along that path)

Ways of Expressing Vector Field Line Integrals

1. $\int \_CF\cdot T$ definition
2. $\int \_CF\cdot dr$ vector differential form
3. $\int \_a^{b}F\cdot \frac{dr}{dt}dt$ parametric vector evaluation
4. $\int \_a^b(M{g}^{\prime }(t)+N{h}^{\prime }(t)+P{k}^{\prime }(t))dt$ parametric scalar evaluation
5. $\int \_a^{b}Mdx+Ndy+Pdz$ scalar differential form