Check if the following fields are conservative and solve the integrals

Question

Let $L$ be the straight section connecting the points $A\left(0,-3,-6\right) \rightarrow B\left(3,2,1\right)$

Solve the following integrals (Check if the fields are conservative to avoid unnecessary work)

$\displaystyle \int_{L} \left( 9 x^2 y^2 z \right) \; dx + \left( 6 x^3 y z + 12 y^2 z \right) \; dy + \left( 3 x^3 y^2 + 4 y^3 \right) \; dz {}$ $\displaystyle \int_{L} \left( 9x^{2}y^{2}z \right) \; dx + \left( 3x+6x^{3}yz+12y^{2}z \right) \; dy + \left( 3x^{3}y^{2}-3y+4y^{3} \right) \; dz {}$

1. What unnecessary work can be avoided if the field is conservative?
2. I started by calculating the curl to check if the field is conservative. In the first integral the curl is 0. Is this suffiecient to say the field is conservative?
3. If the field is not conservative what method do I use to solve the integral?

Thank you.

Answer 1

For conservative vector field, the line integral over a smooth curve starting at point $A$ and ending at point $B$ is path independent and can be evaluated as,

$\displaystyle \int_C \vec{F} \cdot dr = f(B) - f(A) \,$

where $\vec{F} = \nabla f(x, y, z)$ i.e. $\vec{F} \, $ is the gradient of the potential function $f(x, y, z)$.

For your first line integral,

$\displaystyle \int_{L} \left( 9 x^2 y^2 z \right) \; dx + \left( 6 x^3 y z + 12 y^2 z \right) \; dy + \left( 3 x^3 y^2 + 4 y^3 \right) \; dz {}$

The vector field is $\vec{F} = (9 x^2 y^2 z, 6 x^3 y z + 12 y^2 z, 3 x^3 y^2 + 4 y^3)$

As you have already established that the curl is zero, to find the potential function

$f(x, y ,z) = \int F_x dx + g(y, z) = 3x^3y^2z + g(y, z)$

Now if you look at $F_y$, you see the second term as only a function of $y, z$. Integrating that, $g(y, z) = 4y^3z + h(z)$

As there is no term in $F_z$ which is only a function of $z$,

$f(x, y, z) = 3x^3y^2z + 4y^3z$

Your line integral is simply = $f(3, 2, 1) - f(0, -3, -6)$.

While your second vector field is not conservative, you can easily split it into two and take advantage of the work we already did.

$\displaystyle I_2 = I_1 + 3 \int_{L} x \, dy - y \, dz \,$ (if $I_1$ is the line integral from the first problem).

Specifically on your question on parametrization of line segment $AB$,

$r(t) = (0, -3, -6) + t \, (3-0, 2 - (-3), (1 - (-6))$

$ = (0, -3, -6) + (3, 5, 7) t, 0 \leq t \leq 1$

$x = 3t, y = -3 + 5t, z = -6 + 7t$

$dx = 3 \, dt, dy = 5 \, dt, z = 7 \, dt$