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How to avoid a workout (like a lazy mathematician) !
Copyright 2008 by Evans M. Harrell II.
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A different kind of integral in 3D: “line integral”
Work done by a force on a moving object:
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Line integrals
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Line integrals
Important! In a line integral we do not hold x or y fixed while letting the other one vary. !
where F = P i + Q j, and dr = dx i + dy j
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Examples from the previous episode.
In all cases, let the curve be the unit circle, traversed counterclockwise.
1. F(x) = x i + y j
2. F(x) = y i + x j
3. F(x) = y i - x j. This time F · dr =( - sin2 t - cos2 t) dt = - dt. The integral around the whole circle is -2π, even though the
beginning and end points are the same.
Under what conditions is it true that an integral from a to b=a gives us 0 - as in one dimension?
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Under what conditions is it true that an integral from a to b=a gives us 0 - as in one dimension?
Related questions:
1. Is there such a thing as an antiderivative?
2. Does the value of the integral depend on the path you take?
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The fundamental theorem for line integrals, at least some of them…
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The fundamental theorem
Assuming C is ___?____, f is ___?____, and
h = ∇f on a set that is _____?______:
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Path-independence
Can we ever reason that if the curve C goes from a to b, then the integral is just of the form f(b) - f(a), as in one dimension?
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Path-independence
The technicalities of the theorem that path-independence is equivalent to the fact that integrals over all loops are zero. Paths stay within an open, “simply
connected” domain. Curve and vector function F are
sufficiently nice to change variables. Say, continuously differentiable.
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The fundamental theorem
Assuming C is __________, f is ________, and h = ∇f on a set that is ____________:
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Examples
1. F(r) = x i + y j. F(r) = ∇(x2+y2)/2 at each point
2. F(x) = y i + x j. F(r) = ∇ xy
3. F(x) = y i - x j. F(r) is not a gradient.
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, F(r) = y i + x j.
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The fundamental theorem
Assuming C is a piecewise smooth curve, f is continuously differentiable, and
h = ∇f on a set that is open and simply connected:
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The fundamental theorem
Why is this true? Strategy: reduce this question to a one-dimensional integral:
f(r(t)) is a scalar-valued function of one variable. What’s its derivative?
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The fundamental theorem
Why is this true? Strategy: reduce this question to a one-dimensional integral:
f(r(t)) is a scalar-valued function of one variable. What’s its derivative?
∇f(r(t)) · r´(t). By the fundamental theorem of alculus, the integral of this function from t1 to t2 is
f(r(t2)) - f(r(t1)) = f(b) - f(a). Quoth a rat demon, “strand ‘em.”
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The easy way to do line integrals, if h = ∇f
1. h(r) = x i + y j. h(r) = ∇(x2+y2)/2 at each point
2. h(x) = y i + x j. h(r) = ∇ xy
Remind me - how do you find f if h = ∇f?
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A typical example
h(r) = (2 x y3 - 3 x2) i + (3 x2 y2 + 2 y) j Integral would be ∫(2 x y3 - 3 x2)dx + (3 x2 y2 + 2 y)dy 1. Check that h(r) is a gradient. 2. Fix y, integrate P w.r.t. x. 3. Fix x, integrate Q w.r.t. y. 4. Compare and make consistent.
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A typical example
h(r) = (2 x y3 - 3 x2) i + (3 x2 y2 + 2 y) j Py = 6 x y2 = Qx, so we know h = ∇f for
some f. To find f, integrate P in x, treating y as
fixed. We get x2 y3 - x3 + φ, but we don’t really know φ is constant as regards y. It can be any function φ(y) and we still have ∂φ/∂x = 0.
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A typical example
h(r) = (2 x y3 - 3 x2) i + (3 x2 y2 + 2 y) j Now that we know f(x,y) = x2 y3 - x3 + φ, let’s
figure out φ by integrating Q in the variable y: The integral of Q in y, treating x as fixed is
x2 y3 + y2 + ψ, but ψ won’t necessarily be constant as regards x. It can be any function ψ(x) and we still have∂ψ/∂y = 0.
Compare: f(x,y) = x2 y3 - x3 + φ(y) = x2 y3 + y2 + ψ(x) So we can take φ(y) = y2 + C1, ψ(x) = x3 + C2, Conclusion: f(x,y) = x2 y3 - x3 + y2 + C (combining the
two arbitrary constants C1,2 into one).
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The fundamental theorem
Assuming C is a piecewise smooth curve, f is continuously differentiable, and
h = ∇f on a set that is open and simply connected:
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Dern! The pesky little auk up and grabbed the slides from that really cool example done in class and flew off to Baffin Island with ‘em!!
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Conservation of energy
U is the “potential energy.” F is a “conservative force.”
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Conservation of energy
Total energy = kinetic + potential
If r(t) is a curve, then E(t) is a function of t.
In principle.
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Application: Escape veolcity
How fast do you need to blast off to be lost in space? “Ground Control to Major Tom….” !
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