concept of line integral

[janus1609]

I do understand an integral: a summation of f(x)dx , f(x)being the length , dx the infinimestal base of a rectangular.
I do understand a line integral (LI) with respect to arc length ds , so let's say F(x,y)ds

What i fail to understand is that most textbooks go from LI wth respect to ds TO a form like P(x,y)dx + Q(x,y)dy
Without explanation . Not to mention that i do not understand the product of P(x,y)dx or Q(x,y)dy (what are you calculating with P(x,y)dx or Q(x,y)dy
Some books suggest that P(x,y)dx is the projection of the function onto the X-axis , Q(x,y)dy on the Y-axis
And some books seems to suggest that F(x, y)ds can be split into P(x,y)dx + Q(x,y)dy
Again without explaining how to calculate.


Can somebody help me
Do not be annoyed by language mistakes, i'm a foreigner . And I'm not a math teacher, i'm just a math loving amateur who has learned much thanks to youtubue videos of MIT and Yale and the like

 

[HallsofIvy]

I do understand an integral: a summation of f(x)dx , f(x)being the length , dx the infinimestal base of a rectangular.

Unfortunately, what you say you "understand" simply is not true. What you are describing is a "Riemann sum" and an integral is a certain type of limit of a Riemann sum.

I do understand a line integral (LI) with respect to arc length ds , so let's say F(x,y)d

What i fail to understand is that most textbooks go from LI wth respect to ds TO a form like P(x,y)dx + Q(x,y)dy
Without explanation . Not to mention that i do not understand the product of P(x,y)dx or Q(x,y)dy (what are you calculating with P(x,y)dx or Q(x,y)dy
Some books suggest that P(x,y)dx is the projection of the function onto the X-axis , Q(x,y)dy on the Y-axis
And some books seems to suggest that F(x, y)ds can be split into P(x,y)dx + Q(x,y)dy
Again without explaining how to calculate.


Can somebody help me
Do not be annoyed by language mistakes, i'm a foreigner . And I'm not a math teacher, i'm just a math loving amateur who has learned much thanks to youtubue videos of MIT and Yale and the like

There is a difference between the integral of a numeric valued function, \(\displaystyle f(x,y,z)\), along a path and the integral of a vector valued function, \(\displaystyle \vec{f}(x,y,z)\), along a path.

Given a path, in two dimensional space, described by the parametric equations x= f(t), y= g(t), we can write it as a vector valued function: \(\displaystyle \vec{f}(t)= f(t)\vec{i}+ g(t)\vec{j}\). That is the "\(\displaystyle dx\vec{i}+ dy\vec{j}\). Writing the vector valued function \(\displaystyle \vec{f}(x, y)= u(x,y,z)\vec{i}+ v(x,y,z)\vec{j}\), the integration then is the dot product of those vectors, \(\displaystyle \vec{f}\cdot d\vec{s}= u(x,y)dx+ v(x,y)dy\). That is the "P(x,y)dx+ Q(x,y)dy" you are asking about.