Vector Fields and Line Integrals

16.1 Vector Fields

Vector Fields in $ ext{R}^2$ and $ ext{R}^3$
- A vector field in $ ext{R}^2$ is a function that assigns to each point $(x,y) ext{ in D } ext{R}^2$ a 2D vector.
- A vector field in $ ext{R}^3$ assigns to each point $(x,y,z) ext{ in D } ext{R}^3$ a 3D vector:
  F(x,y,z) = (P(x,y,z), Q(x,y,z), R(x,y,z))
- Sometimes $P, Q, R$ are scalar fields.
- Example 1:
  F(x,y) = -y extbf{i} + x extbf{j}
- Here, the vectors at specific points are:
  - $(1,0) o (0,1)$
  - $(2,1) o (1,-2)$
Gradient Fields
- If $f$ is a real-valued function of 2 or 3 variables, then it defines a vector field on $ ext{R}^2$ or $ ext{R}^3$.
- Example:
- If $f(x,y) = x^2y - y^3$, then
  
  F(x,y) =
  egin{pmatrix} rac{ ext{d} f}{ ext{d} x} \ rac{ ext{d} f}{ ext{d} y} \
  =

Arc Length:
- Formula for length of a curve in $ ext{R}^2$:
  L = ext{int}_{a}^{b} ext{sqrt}ig(rac{dx}{dt}^2 + rac{dy}{dt}^2ig) ext{ dt}
- In $ ext{R}^3$, the formula is:
  L = ext{int}_{a}^{b} ext{sqrt}ig(rac{dx}{dt}^2 + rac{dy}{dt}^2 + rac{dz}{dt}^2ig) ext{ dt}
- Review:
- Arc length in $ ext{R}^2$: s(t) = ext{int}_{a}^t ext{sqrt}ig(ig(rac{dx}{dt}ig)^2 + ig(rac{dy}{dt}ig)^2ig) dt
- In $ ext{R}^3$:
  s(t) = ext{int}_{a}^{t} ext{sqrt}ig(ig(rac{dx}{dt}ig)^2 + ig(rac{dy}{dt}ig)^2 + ig(rac{dz}{dt}ig)^2ig) dt
Line Integrals:
- Line integrals are integrals over curves rather than over simple intervals.
- In Physics, if $ds$ is the mass of a wire occupying curve $C$, and its density is given.
- The line integral for a function $f$ along a piecewise smooth curve $C$ is defined using the integral:
- ext{int}{C} f ext{ ds} = ext{lim}igg( ext{summing up } f(pi) ext{ for } i=1 ext{ to } n ext{ increments along the curve}igg)

Example 1: Evaluate ext{int}_{C} (2+x^2y) ext{ ds}
- Where $C$ is the upper half of the unit circle: y^2 = 1 - x^2.
- Step 1: Parametrize the curve:
- $x= ext{cos}(t)$, $y= ext{sin}(t)$, $0 ext{ to } rac{ ext{ ext{π}}}{2}$
- ext{ds} = ext{sqrt}ig( ext{sin}(t)^2 + ext{cos}(t)^2ig) dt.
- Convert the integral into parameters:
- ext{int}_{0}^{2 ext{ ext{π}}} (2 + ext{cos}^2(t) ext{sin}(t)) dt

Let $C$ be an oriented curve in $ ext{R}^2$ or $ ext{R}^3$, with initial point $A$ and terminal point $B$.
Form the partition of $C$ by points $A = P0, P1, ext{and} P_2$ and assume vector field $F$ exists.
The line integral is given as: ext{int}{C} extbf{F} ullet dr = ext{lim}{ ext{part}} igg( ext{summing } extbf{F}(P) ullet dp_i ext{ from } A ext{ to } Bigg)
- Geometrically, the dot product represents the work done by the force field on a particle moving from $A$ to $B$.

The theorem states that if $f$ is continuous on an interval $[a, b]$, the integral from $a$ to $b$ can be represented as: ext{int}_{a}^{b} f(x)dx = F(b) - F(a)
- Where $F$ is any anti-derivative of $f$.

A vector field $F$ is conservative if the line integral $ ext{int}_{C} extbf{F}ullet dr$ does not depend on the path taken from $A$ to $B$.
- For a conservative field, it can be shown that:
  ext{int}_{C} extbf{F} ullet dr = f(r(B)) - f(r(A)).

Curl: The operator curl measures the rotation of the field at a point in space.
- Given vector field F = P $ extbf{i}$ + Q $ extbf{j}$ + R $ extbf{k}$:
- Curl is calculated as:
  ext{curl} F =
  abla imes F = egin{pmatrix}rac{ ext{d}R}{ ext{d}y} - rac{ ext{d}Q}{ ext{d}z}, rac{ ext{d}P}{ ext{d}z} - rac{ ext{d}R}{ ext{d}x}, rac{ ext{d}Q}{ ext{d}x} - rac{ ext{d}P}{ ext{d}y}igg).
- Divergence: The measure of how much a vector field spreads out from a point.
- Calculated as:
  ext{div} F =
  abla ullet F = rac{ ext{d}P}{ ext{d}x} + rac{ ext{d}Q}{ ext{d}y} + rac{ ext{d}R}{ ext{d}z}.