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Introduction to Theoretical Methods Example Differentiation of integrals and Leibniz rule • We are interested in solving problems of the type d Z v(x) f (x; t)dt dx u(x) • Notice in addition to the limits depending on x, the function f (x; t) also depends on x • First lets take the case where the limits are constants (u(x) = b and v(x) = a) • In most instances we can take the derivative inside the integral (as a partial derivative) d Z b Z b @f (x; t) f (x; t)dt = dt dx a a @x Patrick K. Schelling Introduction to Theoretical Methods Example R 1 n −kt2 • Find 0 t e dt for n odd • First solve for n = 1, 1 Z 2 I = te−kt dt 0 • Use u = kt2 and du = 2ktdt, so 1 1 Z 2 1 Z 1 I = te−kt dt = e−udu = 0 2k 0 2k dI R 1 3 −kt2 1 • Next notice dk = 0 −t e dt = − 2k2 • We can take successive derivatives with respect to k, so we find Z 1 2n+1 −kt2 n! t e dt = n+1 0 2k Patrick K. Schelling Introduction to Theoretical Methods Another example... the even n case R 1 n −kt2 • The even n case can be done of 0 t e dt • Take for I in this case (writing in terms of x instead of t) 1 Z 2 I = e−kx dx −∞ • We can write for I 2, 1 1 Z Z 2 2 I 2 = e−k(x +y )dxdy −∞ −∞ • Make a change of variables to polar coordinates, x = r cos θ, y = r sin θ, and x2 + y 2 = r 2 • The dxdy in the integrand becomes rdrdθ (we will explore this more carefully in the next chapter) Patrick K. Schelling Introduction to Theoretical Methods Example: continued • Then in polar coordinates we solve the I 2 integral, 2π 1 Z Z 2 π I 2 = e−kr rdrdθ = 0 0 k p π • So we find I = k 1=2 • To find R 1 x2e−kx2 dx, we take − 1 dI = 1 1 π 0 2 dk 2 2 k3=2 2 1=2 • For R 1 x4e−kx2 dx = 1 d I = 1 1 3 π 0 2 dk2 2 2 2 k5=2 1 1=2 Z 2 π (2n − 1)!! x2ne−kx dx = n+1 (2n+1)=2 0 2 k Patrick K. Schelling Introduction to Theoretical Methods Continuing Leibniz rule... • Remember we wanted to solve problems like d Z v(x) f (x; t)dt dx u(x) • So far have only considered the case u(x) = a and v(x) = b were just constants • Let us now look at the problem when f (x; t) is only a function of t, f (t) dI d Z v(x) = f (t)dt dx dx u(x) • We see I (u; v) and also u(x), v(x) are functions of x, so dI @I du @I dv = + dx @u dx @v dx @I @I • Let's see how we find @u and @v Patrick K. Schelling Introduction to Theoretical Methods Leibniz continued... @I @I • For @u ( @v ) we see we hold v (u) constant Z v I = f (t)dt = F (v) − F (u) u dF (x) • Here we note f (x) = dx @I @I • Hence @v = f (v) and @u = −f (u) • Finally, we find dI dv du = f (v) − f (u) dx dx dx Patrick K. Schelling Introduction to Theoretical Methods Putting it all together... • Remember again we wanted to consider in the integrand f (x; t), and hence we want d Z v(x) f (x; t)dt dx u(x) • We can put together what we found in the preceding slides to find Leibniz rule d Z v(x) dv du Z v @f f (x; t)dt = f (x; v) − f (x; u) + dt dx u(x) dx dx u @x Patrick K. Schelling Introduction to Theoretical Methods Example of Leibniz rule • Solve Section 12, problem 13. Find d Z 2=x sin xt dt dx 1=x t sin xt • We see u(x) = 1=x, v(x) = 2=x and f (x; t) = t du 2 dv 2 @f • We find also dx = −1=x and dx = −2=x and @x = cos xt • Then we can directly use Leibniz rule, d Z 2=x sin xt (sin 1 − sin 2) (sin 1 − sin 2) dt = − = 0 dx 1=x t x x Patrick K. Schelling Introduction to Theoretical Methods Chapter 5: Multiple integrals and applications We often require multiple integrals in physics for obvious reasons. These will include integrals over geometric shapes including line integrals, surface integrals, and volume integrals. Sometimes symmetry and a clever change of variables can simplify multiple integrals to few dimensions. In any case, we need to explore how to use the Jacobian to write integrals in various coordinate systems. Examples include Cartesian, polar, spherical, and cylindrical coordinate systems. Note that the first midterm tests up to the material in chapter 5! (Lecture may go somewhat beyond chapter 5 before the test) Patrick K. Schelling Introduction to Theoretical Methods I Make some applications of multiple integrals to physics problems I Change variables using the Jacobian I Work in typical coordinate systems including Cartesian, polar, cylindrical and spherical I Use surface and volume integrals (line integrals will come in a later chapter) Introduction By the end of chapter 5, you should be able to... I Understand and use double and triple integrals Patrick K. Schelling Introduction to Theoretical Methods I Change variables using the Jacobian I Work in typical coordinate systems including Cartesian, polar, cylindrical and spherical I Use surface and volume integrals (line integrals will come in a later chapter) Introduction By the end of chapter 5, you should be able to... I Understand and use double and triple integrals I Make some applications of multiple integrals to physics problems Patrick K. Schelling Introduction to Theoretical Methods I Work in typical coordinate systems including Cartesian, polar, cylindrical and spherical I Use surface and volume integrals (line integrals will come in a later chapter) Introduction By the end of chapter 5, you should be able to... I Understand and use double and triple integrals I Make some applications of multiple integrals to physics problems I Change variables using the Jacobian Patrick K. Schelling Introduction to Theoretical Methods I Use surface and volume integrals (line integrals will come in a later chapter) Introduction By the end of chapter 5, you should be able to... I Understand and use double and triple integrals I Make some applications of multiple integrals to physics problems I Change variables using the Jacobian I Work in typical coordinate systems including Cartesian, polar, cylindrical and spherical Patrick K. Schelling Introduction to Theoretical Methods Introduction By the end of chapter 5, you should be able to... I Understand and use double and triple integrals I Make some applications of multiple integrals to physics problems I Change variables using the Jacobian I Work in typical coordinate systems including Cartesian, polar, cylindrical and spherical I Use surface and volume integrals (line integrals will come in a later chapter) Patrick K. Schelling Introduction to Theoretical Methods Double and triple integrals • We might want to evaluate a volume integral, for example z = f (x; y) surface to surface bounded by coordinate axes • We have dV = zdxdy, which is just summing up the volume of columns of height z • We have to consider the limits on x and y, where limits might be a function of other variable • For example, find the volume of a solid bounded by z = 1 + y, the vertical plane 2x + y = 2, and the coordinate axes Z 1 Z y=2−2x V = (1 + y)dydx x=0 y=0 • Perform integral on y first, since limits depend on x Z 1 Z 1 V = (2 − 2x) + (2 − 2x)2=2 dx = 4 − 6x + 2x2 dx = 5=3 x=0 x=0 • We could have used x = 1 − y=2, and integrate x first, then y Patrick K. Schelling Introduction to Theoretical Methods Another way... • We observe that our volume is also V = RRR dxdydz and then we just need suitable limits • Could use z = 0 on lower surface, z = 1 + y on upper surface and integrate on z first • Next use y = 0 and y = 2 − 2x on two faces and integrate on y • Lastly x = 0 and x = 1 are limits on x Z x=1 Z y=2−2x Z z=1+y V = dzdydx x=0 y=0 z=0 • After the z integral, we get same thing as before when we were summing up columns, Z 1 Z y=2−2x V = (1 + y)dydx = 5=3 x=0 y=0 Patrick K. Schelling Introduction to Theoretical Methods Volume integrals: coordinate systems • We might want to integrate over a surface or a volume ZZZ F (x; y; z)dxdydz • We might most easily do in another coordinate system, for example spherical coordinates ZZZ F (r; φ, θ)r 2drdφd(cos θ) • Of course d cos θ = − sin θdθ • In general we have dV or dΩ, dV = r 2drdφd cos(θ) • Or in cylindrical coordinates dV = rdrdθdz Patrick K. Schelling Introduction to Theoretical Methods.
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