I am trying to calculate the line integral of a vector field along a given path.

$$\vec{v} = (r \cos^2θ) \hat{{r}} − (r\cosθ\sinθ) \hat{{θ}} + 3r \hat{{φ}}$$

Here, $r$ is the magnitude of the position vector, $\theta$ is the angle of the position vector with respect to the $z$-axis, and $\varphi$ is the angle of the projection in the $xy$ plane with respect to the $x$-axis.

The path is given in the link below:


To compute the integral, I assume we're supposed to first compute it from $(1,0,0)$ to $(0,1,0)$, then from $(0,1,0)$ to $(0,1,2)$ and so on.

I was able to calculate the integral from the point $(1,0,0)$ to $(0,1,0)$, but after that, the vector $\vec{r}$ is no longer of constant magnitude, so I do not know how to proceed. Do I have to convert into Cartesian coordinates, or is it possible to still solve this using spherical polar coordinates?

This is what I did to calculate the line integral from $(1,0,0)$ to $(0,1,0)$:

First, I wrote the unit vectors $\hat{{r}}$, $\hat{{θ}}$ and $\hat{{φ}}$ in terms of $\hat{{i}}$, $\hat{{j}}$, $\hat{{k}}$

$$\hat{{r}} = \cos\varphi \hat{{i}} + \sin \varphi \hat{{j}}$$

$$\hat{{\theta}} = - \hat{{k}}$$

$$\hat{{\varphi}} = - \sin\varphi \hat{{i}} + \cos\varphi \hat{{j}}$$

Now, substituting these in $\vec{v}$, I got

$$\vec{v} = (r cos^2\theta \cos\varphi - 3r \sin\varphi) \hat{{i}} + (r \cos^2\theta \sin\varphi + 3r \cos\varphi) \hat{{j}} + (r \cos\theta \sin\theta) \hat{{k}}$$

Also, $dr = -\sin\varphi d\varphi \hat{{i}} + \cos\varphi d\varphi \hat{{j}}$

Taking the dot product, I got $\vec{v}\cdot dr = 3r \cos^2\varphi d\varphi$

On integrating this from $\varphi=0$ to $\varphi=\pi/2$, I ended up with $0$.

But now, to go from $(0,1,0)$ to $(0,1,2)$, I am stuck, as this is no longer a circular/spherical path, and the magnitude of $r$ keeps changing. I suppose I could write $\vec{r} = \hat{{j}} + z \hat{{k}}$, but I do not know how to write the expressions of the unit vectors $\hat{{\theta}}$ and $\hat{{\varphi}}$ in terms of $\hat{{i}}$, $\hat{{j}}$, $\hat{{k}}$.

Any help would be greatly appreciated!


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