Graphical illustration of the relationship between curve integrals of the first kind and curve integrals of the second kind

Graphical illustration of the relationship between curve integrals of the first kind and curve integrals of the second kind

Notes related content:
1. Line Integral
2. Line Integral, Circulation, and Flux in Vector Fields

Curve integral of the first kind (for arc length $ds$ for integration)(no directionality)
Physical meaning:$f(x,y)$ can be the linear density,$ds$ is the arc length element, in the curve$Integrate\; over\; L$ to get the curve quality
$${\int }_{L}f(x,y)ds$$
second kind of curve integral (for coordinates$dx,dy$ are integrated)(directional)
Physical meaning:$\vec{F}$is a variable force, $d\vec{r}$is the displacement element, along the curve LLIntegrate in a certain direction of$L$
to get the work W d W = F ⃗ ⋅ dr ⃗ d W = { P ( x , y ) , Q ( x , y ) } ⋅ { dx , dy } d W = P ( x , y ) dx + Q ( x , y ) dy ∫ L d W = ∫ LF ⃗ ⋅ dr ⃗ = ∫ LP ( x , y ) dx + Q ( x , y ) dy dW=\vec{ F}\cdot d\vec{r}\\ ~\\ dW=\{P(x,y),Q(x,y)\}\cdot\{dx,dy\}\\ ~\\ dW= P(x,y)dx+Q(x,y)dy\\ ~\\ \int_LdW=\int_L\vec{F}\cdot d\vec{r}=\int_LP(x,y)dx+Q( x,y)dydW=F ⋅dr  dW={ P(x,y),Q(x,y)}⋅{ dx,dy} dW=P(x,y)dx+Q(x,y ) d y ∫L​dW=∫L​F ⋅dr =∫L​P(x,y)dx+Q(x,y)dy
其中
$P(x,y)$ is the variable force$\vec{F}$At point $(x,y)$ alongxxComponent Q ( x , y ) Q(x,y)$in\; the\; x-$ axis direction
$Q(x,y)$ is the variable force$\vec{F}$At point $(x,y)$ alongyyComponent force dx dx$in\; the\; y-$ axis direction
$dx$ is at point$(x,y)$ alongxxSmall displacement dy dy$in\; the\; x-$ axis direction
$dy$ is at point$(x,y)$ along$Small\; displacement\; in\; the\; y-$ axis direction
$P(x,y)dx$ is the variable force$\vec{F}$At point $(x,y)$ alongxxWork Q ( x , y ) dyof small displacement in the$x-\; axis\; direction\; Q(x,y)dy$
$Q(x,y)dy$ is the variable force$\vec{F}$At point $(x,y)$ alongyyThe work of small displacement in$the\; y-\; axis\; direction$

The relationship between curve integrals of the first kind and curve integrals of the second kind


$${\int }_{L}P(x,y)dx+Q(x,y)dy={\int }_{L}P(x,y)\cos \alpha ds+Q(x,y)\cos \beta ds$$