5.5 Path-Independent Vector Fields and the Fundamental Theorem of Calculus for Line Integrals

Path-Independent Vector Fields and the Fundamental Theorem of Calculus for Line Integrals

Introduction

Path-Independent Vector Fields

In Activity 5.4.3, we considered the vector field $\vF(x,y) = \langle y^2,2xy+3\rangle$ and two different oriented curves from $(-2,5)$ to $(3,30)$. We found that the value of the line integral of $\vF$ was the same along those two oriented curves.

Verify that $\vF(x,y) = \langle y^2,2xy+3\rangle$ is a gradient vector field by showing that $\vF = \nabla f$ for the function $f(x,y) = xy^2 + 3y$.

Work through this exercise and explain your reasoning step by step.

$\int_C \nabla f\cdot d\vr$ if $f(x,y) = 3xy^2 - \sin(x) + e^y$ and $C$ is the top half of the unit circle oriented from $(-1,0)$ to $(1,0)$.

$\int_C \nabla g\cdot d\vr$ if $g(x,y,z) = xz^2 - 5y^3\cos(z) + 6$ and $C$ is the portion of the helix $\vr(t) = \langle 5\cos(t),5\sin(t),3t\rangle$ from $(5,0,0)$ to $(0,5,9\pi/2)$.

$\int_C \nabla h\cdot d\vr$ if $h(x,y,z) = 3y^2e^{y^3} - 5x\sin(x^3z) + z^2$ and $C$ is the curve consisting of the line segment from $(0,0,0)$ to $(1,1,1)$, followed by the line segment from $(1,1,1)$ to $(-1,3,-2)$, followed by the line segment from $(-1,3,-2)$ to $(0,0,10)$.

If $\vG$ and $\vH$ are to be gradient vector fields, then there are functions $g$ and $h$ for which $\vG = \nabla g$ and $\vH=\nabla h$. If such functions $g$ and $h$ exist, what would $g_y$, $g_z$, $h_x$, $h_y$, and $h_z$ be?

Let $g_1(x,y,z)=3xe^{y^2}+xyz-z\sin(x)$. Calculate $\partial g_1/\partial x$. Could $g_1$ be a potential function for the vector field $\vG$?

Find a function $g$ so that $\partial g/\partial x = 3e^{y^2}+z\sin(x)$. Find a function $h$ so that $\partial h/\partial x = 3x^2y$.

Let $\vG(x,y,z) = \langle 3e^{y^2}+z\sin(x),6xy e^{y^2} - z,3z^2-y-\cos(x)\rangle$ and $\vH(x,y,z) = \langle 3x^2 y,x^3+2yz^3,xz+3y^2z^2\rangle$.

Now calculate $\partial g/\partial y$ and $\partial h/\partial y$ based on your choices for . Write a few sentences to explain why this tells you that we must have

and

for some functions $m_1$ and $m_2$ depending only on $z$.

Calculate $\frac{\partial g}{\partial z}$ and $\frac{\partial h}{\partial z}$ for the functions in the part above. Notice that $m_1$ and $m_2$ are functions of $z$ alone, so taking a partial derivative with respect to $z$ is the same as taking an ordinary derivative, and thus you may use the notation $m'_1(z)$ and $m'_2(z)$.

Explain why $\vG$ is a gradient vector field but $\vH$ is not a gradient vector field. Find a potential function for $\vG$.

$\int_C \vF\cdot d\vr$ if $\vF(x,y) = \langle 2x,2y\rangle$ and $C$ is the line segment from $(1,2)$ to $(-1,0)$.

$\int_C \vG\cdot d\vr$ if $\vG(x,y) = \langle 4x^3-12y\cos(xy),9y^2-12x\cos(xy)\rangle$ and $C$ is the portion of the unit circle from $(0,-1)$ to $(0,1)$.

$\int_C \vH\cdot d\vr$ if $\vH(x,y,z) = \langle H_1,H_2,H_3\rangle$ with

and $C$ is the curve consisting of the line segment from $(1,1,1)$ to $(3,0,3)$, followed by the line segment from $(3,0,3)$ to $(1,5,-1)$, followed by the line segment from $(1,5,-1)$ to $(0,0,0)$.