[6.F] Practice vector line integrals

We have two equivalent ways to calculate vector line integrals: $$\int_C \myv F\cdot d\myv r = \int_{t=a}^b\myv F(t)\cdot \myv r'(t)\,dt$$ Writing $F(x,y)=P\,\uv{\imath}+Q\,\uv{\jmath}+R\,\uv{k}$ and $d\myv r=dx\,\uv{\imath}+dy\,\uv{\jmath}+dz\.\uv{k}$, the first formula becomes $$\int_CP\,dx+\int_CQ\,dy+\int_CR\,dz $$ This is particularly useful if you're integrating along a path, $C$, which is parallel to one of the Cartesian directions!

The second formula requires that you find a parameterization $\myv r(t)$ for the path of integration. Then you must also calculate $\myv r'(t)$ and express $x(t)$, $y(t)$, and $z(t)$ as functions of the parameter $t$ along your path.

Practice

Consider two paths going from $A=(2,1)$ to $B=(2,3)$. For each of the vector fields $\myv F(x,y)$ below, calculate the vector line integrals $\int_I \myv F\cdot d\vec{\bf r}$ and $\int_{II+III} \myv F\cdot d\vec{\bf r}$.

Additionally, if the vector field passes the curl test, $\myv \grad\times \myv F=Q_x-P_y=0$, find the potential function $\phi(x,y)$ that fulfills $\myv \grad \phi=\myv F$, and calculate the integral by means of $\int_A^B\myv F\cdot d\myv r=\phi(B)-\phi(A)$ (independent of path).

1. $\myv F=P\uv{\imath}+Q\uv{\jmath} = y\uv{\imath}+(3x+y)\uv{\jmath}$
2. $\myv F=P\uv{\imath}+Q\uv{\jmath} = y\uv{\imath}+(x+y)\uv{\jmath}$