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} \
    =

16.2 Line Integrals

• Arc Length:
  • Formula for length of a curve in $ ext{R}^2$:
    $L = ext{int}_{a}^{b} ext{sqrt}\big(\frac{dx}{dt}^2 + \frac{dy}{dt}^2\big) ext{ dt}$
  • In $ ext{R}^3$, the formula is:
    $L = ext{int}_{a}^{b} ext{sqrt}\big(\frac{dx}{dt}^2 + \frac{dy}{dt}^2 + \frac{dz}{dt}^2\big) ext{ dt}$
  • Review:
  • Arc length in $ ext{R}^2$: $s(t) = ext{int}_{a}^t ext{sqrt}\big(\big(\frac{dx}{dt}\big)^2 + \big(\frac{dy}{dt}\big)^2\big) dt$
  • In $ ext{R}^3$:
    $s(t) = ext{int}_{a}^{t} ext{sqrt}\big(\big(\frac{dx}{dt}\big)^2 + \big(\frac{dy}{dt}\big)^2 + \big(\frac{dz}{dt}\big)^2\big) dt$
• 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:
  • $ext{int}<em>{C} f ext{ ds} = ext{lim}\bigg( ext{summing up } f(p</em>i) ext{ for } i=1 ext{ to } n ext{ increments along the curve}\bigg)$

Example Evaluations

• 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}
  • $ext{ds} = ext{sqrt}\big( ext{sin}(t)^2 + ext{cos}(t)^2\big) dt$.
  • Convert the integral into parameters:
  • ext{int}_{0}^{2 ext{ ext{π}}} (2 + ext{cos}^2(t) ext{sin}(t)) dt

Evaluation of Line Integrals with Respect to Coordinates

• 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 $A = P<em>0, P</em>1, ext{and} P_2$ and assume vector field $F$ exists.
• The line integral is given as: $ext{int}<em>{C} extbf{F} \bullet dr = ext{lim}</em>{ ext{part}} \bigg( ext{summing } extbf{F}(P) \bullet dp_i ext{ from } A ext{ to } B\bigg)$
  • Geometrically, the dot product represents the work done by the force field on a particle moving from $A$ to $B$.

Fundamental Theorem for Line Integrals

• 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 $f$.

Conservative Fields

• 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:
    $ext{int}_{C} extbf{F} \bullet dr = f(r(B)) - f(r(A))$.

Curl and Divergence

• 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:
    $ext{div} F = <br />\nabla \bullet F = \frac{ ext{d}P}{ ext{d}x} + \frac{ ext{d}Q}{ ext{d}y} + \frac{ ext{d}R}{ ext{d}z}$.