- #1

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## Homework Statement

Given a scalar function ##f## find (a) ##∫f \vec {dl}## and (b) ##∫fdl##

along a straight line from ##(0, 0, 0)## to ##(1, 1, 0)##.

## Homework Equations

##f(x,y,z) = 12xy + z##

##\vec {dl} = (\vec {dx},\vec {dy},\vec {dz})##

## The Attempt at a Solution

[/B]

So what I'm mainly confused about is with part a, I can't seem to understand what it's referring to thus don't know how to go about starting the integral.

Is it implying that I need to parametricize the scalar function and then take the integral wrt to dl, treating it like a vector valued function approaching it with the dot product like this $$u = f(x,y,z) = 12xy + z , v = y , w = z $$ resulting in the vector function $$\vec {r}(u,v,w) = (12xy + z) \hat {\mathbf i} + y \hat {\mathbf j} + z \hat {\mathbf k} $$ with the parametrization of $$\vec {r}(t) = (t,t,0)$$ resulting in this integral $$\int (12t^2,t,0)⋅(1,1,0) dt$$

or

am I overthinking this and I'm simply looking find the line integral over each of the differential components like this: $$\int f \vec{dl} = \int f {\mathbf dx} \hat {\mathbf i} + \int f {\mathbf dy} \hat {\mathbf j} + \int f {\mathbf dz} \hat {\mathbf k}$$