Let's say there's a creek.

Down in the creek, there is a current.

The vector field that describes this current is

Cur[x y z] = [(πy)^x]i + [(y⁴)+(xyz)]j + (2z + e^z)k

I guess this force can just be expressed in Newtons.


X, Y, and Z are the spatial dimensions in meters, whose origin is a piece of bait in this case.
You see, there's also this guy who's fishin' in the creek. His bait's down there, situated on the origin.

A fish sees it, and circles around it one complete time. The fish is unsure during this period, and maintains a distance of one meter.

This motion takes exactly 2π seconds for the fish.
So, the motion can be described as a vector function of t, time (sec.)

Fis(t) = [cos(t)]i + [sin(t)]j + [0]k





How much energy is used or maintained by the fish?






Ok... so I made this problem up... that's why it's so weird. :)


I need help setting it up. I know I need to use a line integral.

The upper limit, t, in seconds, will be 2π, while the lower will obviously be 0.

So, first off, I need to find the integrand, which is the dot product Cur[Fis(t)] and Fis'(t).

To begin

Cur[Fis(t)] = [(πsin(t))^cos(t)]i + [sin(t)⁴]j + 1k




But here's some trouble for me...
Tell me, O somebody-who-is-doubtlessly-wiser-than-I, would

Fis'(t) = [-sin(t)]i + [cos(t)]j + 0k ?


If it is, I'll continue to find the dot product, and then begin the actual integration.




EDIT: π is pi.

Since I'm bored and no one seems willing to help right now, I'll continue the calculation, whether or not it's right so far.


Cur[Fis(t)] = [(πsin(t))^cos(t)]i + [sin⁴(t)]j + 1k


Fis'(t) = [-sin(t)]i + [cos(t)]j + 0k




Therefore,

Cur[Fis(t)] * Fis'(t) = -sin(t)(πsin(t))^cos(t) + cos(t)sin⁴(t)



Edit:

So...

∫ -sin(t)(πsin(t))^cos(t) + cos(t)sin⁴(t) dt


Integrating that is going to be a pain...

Let's say there's a creek.

Down in the creek, there is a current.

The vector field that describes this current is

Cur[x y z]) = [(πy)^x]i + [(y⁴)+(xyz)]j + (2z + e^z)k


You were in trouble at this point when you let the fish swim around the origin. When y is negative, what is (πy)^x? You current is complex valued. What kind of river is this :bugeye:?

Damn... you're right. The force vector function along the curve Fis has an imaginary i component part of the time.

The 'force * velocity' function(t) isn't continuous, due to that.

Well... it's continuous on the domain (0,pi).
Hmmm...


With the help of my computer, I can integrate on that domain, and I'll then determine half of the work the fish does, I guess.
It computer says approximately -2.1927, which is correct as far as I can tell.

During the first half of his journey, the fish is helped by the current. He is helped 2.1927 Newton-meters.


Edit:

Now... is there any way to get the rest of the work?