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Unformatted text preview: integral along with We write these in their differential form The vectors along the path are of the form: The tangent vector along the curve is: Note that
((we just showed that with the line integral)) But what does it
mean? That the Force Vectors are ALWAYS perpendicular to the path of motion and therefore NO WORK IS DONE! You may have noticed that we can compact the process of the above example with some nice notation: Can work be negative? YES. Remember, all that means is that the force field impedes movement along the path.
The beauty of notation: It looks like the ds’s just cancel each other out. There is no algebra here. That is NOT what is happening. Instead, our
notation is such that these nice pneumonic devices occur all the time in calculus.
This is just the method of substitution for an integral which is sort of like the integration version of the chain rule.
Let and let x be some function of s: Chain Rule:
Integrate both sides with respect to s: Since
Putting the two together: Along those lines of notation: (which we showed earlier with a lot more notation) In general, does using a different path lead to a different value for our line integral? YES.
Evaluate where C is the arc of the parabola in the plane from A(0,0,2) to B(1,1,2).
Instead of using t as my parameter I’m going to use the variable x. Thus I must convert dy and dz and substitute for y
and z in the integral. dy = 2x dx and dz = 0 dx Also notice that x goes from 0 to 1 along the curve C.
Now we will use a different path but the same endpoints C: in the plane A(0,0,2) to B(1,1,2). dy = dx and dz = 0 dx What would happen if we used the same path, the same endpoints, but changed the orientation of the curve?
All that would happen here is the integration limits would change:
Evaluation Theorems: Summary for 14.2:
Evaluate a line integral with respect to arc length, x, y, and z.
Evaluate a line integral over a piecewise-smooth curve
Know the affect of orientation on line integrals
Compute work done by a force field Opposite Sign! Section 14.3 Independence of Path and Conservative Vector Field...
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