Since it is just a sort of discussion, I will just give the formula and condition without proving them or leaving examples.
$\int_{C}\vec{F}\cdot \mathrm{d}\vec{r} = \int_{C}M\mathrm{d}x+N\mathrm{d}y$, in which $\vec{F} =
Method: Express $x$ and $y$ in a single variable (OR means parameterization).
$curl(\vec{F}) = 0$ and $\vec{F}$ is defined in a simple-connected region, in which $curl(\vec{F}) = N_{x} - M_{y}$ if $\vec{F} =
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
P & Q & R
\end{vmatrix}
) if $\vec{F} =
$
then $vec{F} = \nabla f$, or $vec{F}$ is the partial derivative vector of some vector field.
Method 1. Do line integral. Integral along the x-axis and y-axis and z-axis, if they exist. (Using path-independence)
Method 2. Integral one component of $\vec{F}$ and then differential it over another variable and compare. (…)
$\hat{n} = \hat{T}$ rotated 90 degrees clockwise $=<\mathrm{d}y,-\mathrm{d}x>$
$\int_{C}\vec{F}\cdot\hat{n}\mathrm{d}s = \int_{C}P\mathrm{d}y-Q\mathrm{d}x$, in which $\vec{F} =
$
$\iint_{S}\vec{F}\cdot\hat{n}\mathrm{d}S = \iint_{S}\vec{F}\cdot(<-f_{x},-f_{y},1>\mathrm{d}x\mathrm{d}y)$, if we use $z = f(x,y)$ to describe the surface.
$=\iint_{S}\vec{F}\cdot(\pm\frac{\vec{N}}{\vec{N}\cdot\hat{k}}\mathrm{d}x\mathrm{d}y)$, if we are given the normal vector of the surface,or specifically, $g(x,y,z) = 0$
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