Lecture 18.5      

More Line Integrals

APMA E2000

Drew Youngren dcy2@columbia.edu

$\gdef\RR{\mathbb{R}}$ $\gdef\vec{\mathbf}$ $\gdef\bv#1{\begin{bmatrix} #1 \end{bmatrix}}$ $\gdef\proj{\operatorname{proj}}$ $\gdef\comp{\operatorname{comp}}$

Announcements

    • No quiz this week
    • HW 11 due next Tues
    • Recitation as usual

Alternate Forms

In Components
$\vec F(x,y) = \left\langle P(x,y), Q(x,y) \right\rangle$
and $\vec r(t) = \langle x(t), y(t) \rangle, a \leq t \leq b$
\[\int\limits_{\mathcal C} \vec F\cdot d\vec r = \int_a^b \langle P(x(t),y(t)), Q(x(t),y(t)) \rangle\cdot \langle x'(t), y'(t)\rangle \,dt\]
\[ = \int_a^b P(x(t),y(t)) \, x'(t)\,dt + Q(x(t),y(t))\, y'(t) \,dt\]
\[ = \int\limits_{\mathcal C} P\,dx + Q\,dy\]
Per Arc Length
\[\int\limits_{\mathcal C} \vec F\cdot d\vec r = \int_a^b \vec F(\vec r(t))\cdot \vec r'(t) \,dt\]
\[= \int_a^b \vec F(\vec r(t))\cdot \frac{ \vec r'(t) }{|\vec r'(t)|} |\vec r'(t)| \,dt\]
\[ = \int\limits_{\mathcal C} \vec F \cdot \vec T\,ds \]
Example

$C$ is the triangle connecting $(-1, 0)$, $(1,1)$, and $(1,0)$.
Order the following line integrals from least to greatest.

$\int_C y\,dx$
$\int_C x\,dx$
$\int_C x\,dy$
$\int_C y\,dx - x\,dy$

Conservative Fields

Exercise

Which vector field below is conservative?

  1. $x\,\mathbf i$
  2. $x\,\mathbf j$
  3. both of these
  4. neither of these
Example

Compute the line integral \[\int_C (x^2 - xy)\,dx + (y- \frac{x^2}{2} + 2)\,dy \] where $C$ is the polygonal path from $(2,0)$ to $(0,0)$ to $(2,1)$ to $(0,1)$.

Z-shaped path in a vector field
Conservation of Energy

Suppose a force is given by $\vec F = -\nabla f$ and a particle of mass $m$ traverses a path $\vec r(t), a \leq t \leq b$.

\[ \text{W} = \int\limits_C \vec F\cdot d\vec r\]
Conservation of Energy

Suppose a force is given by $\vec F = -\nabla f$ and a particle of mass $m$ traverses a path $\vec r(t), a \leq t \leq b$.

\[ \text{W} = \int\limits_C \vec F\cdot d\vec r = f(\vec r(a)) - f(\vec r(b))\]
Conservation of Energy

Suppose a force is given by $\vec F = -\nabla f$ and a particle of mass $m$ traverses a path $\vec r(t), a \leq t \leq b$.

\[ \begin{align*}\text{W} &= \int_a^b m\vec r''(t)\cdot \vec r'(t)\,dt \\ &= \int_a^b \frac{d}{dt}\left(\frac12 m \vec r'(t) \cdot \vec r'(t) \right)\,dt \\ &= \frac12 m |\vec r'(b)|^2 - \frac12 m |\vec r'(a)|^2 \end{align*}\]
Conservation of Energy

Suppose a force is given by $\vec F = -\nabla f$ and a particle of mass $m$ traverses a path $\vec r(t), a \leq t \leq b$.

\[ \frac12 m |\vec r'(b)|^2 + f(\vec r(b)) = \frac12 m |\vec r'(a)|^2 + f(\vec r(a)) \]
Escape Velocity

The Gravitational force from a mass $M$ at the origin on a mass $m$ at position $\vec x$ is \[ \vec G = -\frac{GMm \vec x}{|\vec x|^3}. \] Find the work done moving an object from the surface of the earth to the end of the universe.

On Earth
\[\begin{align*}G &= 6.674 \times 10^{-11} \frac{\text{N}\text{m}^2}{\text{kg}^2} \\ M &= 5.97\times10^{24} \text{kg} \\ R &= 6.371\times 10^{6} \text{m} \end{align*}\] Escape velocity is $\sqrt{\frac{2GM}{R}} = 11183.9 \frac{\text{m}}{\text{s}}$.

Learning Outcomes

You should be able to...
  • Interpret a line integral of a vector field as sum of line integrals of scalar fields.
  • Determine if a vector field is conservative and find a potential if so.
  • Recognize when and how to apply FTLI.
  • Equate path-independence with closed loops yielding 0.
  • Solve mechanical problems with conservative forces.