$ \int_{AB} yz dx + zx dy + xy dz $
This is my line integral and I need to find out a method to solve it. I have 2 points A(1,1,0) and B(2,3,1). These 2 points form a line in space. What I find hard to comprehend is that I have to apply this theory:
$ \frac {\partial P}{\partial y} = \frac{\partial Q}{\partial x} $
$ \frac {\partial Q}{\partial z} = \frac{\partial R}{\partial y} $
$ \frac{\partial R}{\partial x} = \frac{\partial P}{\partial z} $
What do these partial derivatives represent? Functions?
Besides that,there is another formula shown below that represent 3 integrals. These 3 integrals represent a function in space? If so,what does the variable $t$ means? A constant?
$V(x,y,z)= \int_{x_0}^{x} P(t,y_0,z_0) dt+\int_{y_0}^{y} Q(x,t,z_0)dt+\int_{z_0}^{z}R(x,y,t)dt$
Can someone explain each part to me? I would appreciate your help.
The notation $\int_C P \, dx + Q \, dy + R \, dz$ is also sometimes written as $\int_C \vec{F} \cdot d \vec{r}$, where $\vec{F} = \left< P,Q,R \right>$, so $P,Q,R$ are the components of the vector field. In your example, $P = yz$, $Q = zx$, and $R = xy$.
The equations you wrote with the partial derivatives are another way of saying that the "curl" of this vector field is zero. In math notation, $\nabla \times \vec{F} = \vec{0}$. I assume you've seen curl before? This means the vector field is "conservative".
When a vector field is conservative, it means there is a "potential function" $f(x,y,z)$, so that $\vec{F}$ is its gradient. In other words, $\vec{F} = \nabla f$.
The Fundamental Theorem of Line Integrals says that the value of this line integral is the difference of the values of the potential function at the endpoints: $$ \int_{AB} \vec{F} \cdot d \vec{r} = f(B) - f(A) $$
So if you can find this potential function $f$ (I hope you can do that yourself, it's really quite easy in this example), then you don't need to do the integral at all! You just plug the points $A$ and $B$ into this function and subtract.