How to avoid a workout (like a lazy mathematician)!Copyright 2008 by Evans M. Harrell II.A different kind of integral in 3D: “line integral”  Work done by a force on a moving object:Line integralsLine 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 jExamples 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?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?The fundamental theorem for line integrals, at least some of them…!!!!!The fundamental theorem  Assuming C is ___?____, f is ___?____, and h = ∇f on a set that is _____?______: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?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.The fundamental theorem  Assuming C is __________, f is ________, and h = ∇f on a set that is ____________: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., F(r) = y i + x j.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: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?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.”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?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.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.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).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: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!!Conservation of energy U is the “potential energy.” F is a “conservative force.”Conservation of energy Total energy = kinetic + potential If r(t) is a curve, then E(t) is a function of t. In principle.Application: Escape veolcity How fast do you need to blast off to be lost in space? “Ground Control to Major