CAS PROJECT. Graphing Potential Fields. (a) Graph

Find and sketch the potential. Find the complex potential:Between parallel plates at x = - 3 and 3. potentials 140 V and 260 V. respectively

Find and sketch the potential. Find the complex potential:Between parallel plates at t = -4 and 10. potentials 4.4 kV and 10 kV. respectively

Find and sketch the potential. Find the complex potential:Between the axes (potential 110 V) and the hyperbola xy = I (potential 60 V)

Find and sketch the potential. Find the complex potential: Between parallel plates at y = x and x + k. potentials o and 100 V. respectively

Find the potential between two infinite coaxial cylinders of radii r1 and rz having potentials VI and Vz, respectively. Find the complex potential.rl = 0.5, rz = 2.0, VI = -IIOV. V2 = llOV

Find the potential between two infinite coaxial cylinders of radii r1 and rz having potentials VI and Vz, respectively. Find the complex potential.1'1 = 1, r2 = 10. VI = 100 V. V2 = I kV

Find the potential between two infinite coaxial cylinders of radii r1 and rz having potentials VI and Vz, respectively. Find the complex potential. 1'1 = I, 1"z = 4. VI = 200 V. V2 = 0

Find the potential between two infinite coaxial cylinders of radii r1 and rz having potentials VI and Vz, respectively. Find the complex potential. 1'1 = 0.1. 1'2 = 10. VI = 150 V. V2 = 50 V

Show that <1> = el'Tr = (l/'Tr) arctan (ylx) is harmonic in the upper half-plane and satisfies the boundary condition <1>(x, 0) = I if x < 0 and 0 if x > O. and the cOlTesponding complex potential is F(z) = -(i/'Tr) Ln z.

Map the upper half z-plane onto the unit disk Iwl ~ I so that 0, x. - I are mapped onto I, i, -i, respectively. What are the boundary conditions on Iwl = I resulting from the potential in Prob. 9'! What is the potential at w = O?

Verify by calculation that the equipotential lines in Example 7 are circles

CAS EXPERIMENT. Complex Potentials. Graph the equipotential lines and lines offorce in (a)-(d) (four graphs, Re F(z) and 1m F(z) on the same axes). Then explore further complex potentials of your choice with the purpose of discovering configurations that might be of practical interest. (a) F(z) = :;:2 (e) F(z) = liz (b) F(z) = iz 2 (d) F(:;:) = ifz

Show that F(z) = arccos z (defined in Problem Set 13.7) gives the potential in Figs. 398 and 399. Fig

Find the real and complex potentials in the sector -'Tr16 ~ e ~ 'Tr16 between the boundary e = 'Tr16 (kept at 0) and the curve X3 - 3xy2 = I, kept at 110 V.

Find the potential in the first quadrant of the x)'-plane between the axes (having potential 220 V) and the hyperbola xy = I (having potential 110 V).

Verify Theorem 1 for <1>':'(11. U) = 112 - U2 W = fez) = eZ and any domain D.

Verify Theorem I for <1>*(lI. u) = lIU, W = .Hz) = eZ and D: x ::::2 0,0::::2 Y ::::2 7T, Sketch D and D*

Carry out all steps of the second proof of Theorem I (given in App. 4) in detaiL

Derive (3) from (2).

Let D'~ be the image of the rectangle D: o ~ x ::::2 ~ 7T, 0 ::::2 Y ::::2 1 under IV = sin z, and <1>*(11, U) = 112 - u2 Find the corresponding potential <1> in D and its boundary values.

What happens in Prob. 5 if you replace the potential by the conjugate <1>* = 2ltu? Sketch or graph some of the equipotential lines <1> = canst

CAS PROJECT. Graphing Potential Fields. (a) Graph equipotential lines in Probs. 1 and 2. (b) Graph equipOtential lines if the complex potential is F(z) = i;:.2. F(z) = eZ F(;:.) = iez, F(z) = eiz. (c) Graph equipotential surfaces corresponding to F(z) = In z as cylinders in space.

TEAM PROJECT. Noncoaxial Cylinders. Find the potential between the cylinders C1 : Izl = I (potential VI = 0) and C2 : Iz - cl = c (V2 = 110 V), where o < c < ~. Sketch or graph the equipotential curves and their orthogonal trajectories for c = 0.1, 0.2, 0.3, OA. Try to think of the further extension C1: Izi = 1, C2 : I::: - cI = p *- c.

Find the potential <1> in the region R in the first quadrant of the z-plane bounded by the axes (having potential VI) and the hyperbola y = l/x (having potential 0) in two ways. (i) directly, (ii) by mapping R onto a suitable infinite strip,

(Extension of Example 2) Find the linear fractional transfonnation z = g(Z) that maps IZI ::::2 I onto Izi ~ 1 with Z = il2 being mapped onto Z = O. Show that ZI = 0.6 + 0.8i is mapped onto z = -I and Z2 = -0.6 + 0.8i onto::: = I, so that the equipotential lines of Example 2 look in Izi ::::2 1 as shown in Fig. 402.

The equipotential lines in Prob. 10 are circles. Why?

Show that in Example 2 the .v-axis is mapped onto the unit circle in the w-plane.

Find the complex and real potentials in the upper half-plane with boundary values 0 if x < 4 and 10 kV if x > 4 on the x-axis.

(Angular region) Applying a suitable conformal mapping. obtain from Fig. 401 b the potential <1> in the angular region -!7T < Arg z < !7T such that = - 3 kV if Arg z = -!7T and <1> = 3 kV if Arg z = !7T.

At z = 1 in Fig. 401 a the tangents to the equipotential lines shown make equal angles (7T/6). Why?

CAS PROJECT. Isotherms. Graph isothenns and lines of heat flow in Examples 2-4. Can you see from the graphs where the heat flow is very rapid?

Find the temperature and the complex potential in an infinite plate with edges y = x - 2 and y = x + 2 kept at - 10C and 20De, respectively

Find the temperature between two parallel plates \" = 0 and )" = d kept at temperatures ODC and 100De, respectively. (i) Proceed directly. (ii) Use Example I and a suitable mapping.

Find the temperature T in the sector 0 ~ Arg z ~ w/3, Izl ~ 1 if T = 20C on the x-axis, T = 50C on y = V3 x, and the curved portion is insulated.

Find the temperature in Fig. 405 if T = -20DC on the y-axis, T = 100DC on the x-axis, and the circular portion of the boundary is insulated as before.

Interpret Prob. 10 in Sec. 18.2 as a heat flow problem (with boundary temperatures, say, 20DC and 300C). Along what curves does the heat flow?

Find the temperature and the complex potential in the first quadrant of the ;:-plane if the y-axis is kept at 100C, the segment 0 < x < I of the x-axis is insulated and the portion x > I of the x-axis is kept at 200De. Hint. Use Example 4.

TEAM PROJECT. Piecewise Constant Boundary Temperatures. (a) A basic building block is shown in Fig. 407. Find the corresponding temperature and complex potential in the upper half-plane. (b) Conformal mapping. What temperature in the first quadrant of the ;:-plane is obtained from la) by the mapping w = a + Z2 and what are the transfonned boundary conditions?(c) Superposition. Find the temperature T'" and the complex potential F* in the upper half-plane satisfying the boundary condition in Fig. 408. (d) Semi-infinite strip. Applying H' = cosh;: to (c), obtain the solution of the boundary value problem in Fig. 409.

Find the temperature T(x, y) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are insulated as shown.

Find the temperature T(x, y) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are insulated as shown.

Find the temperature T(x, y) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are insulated as shown.

Find the temperature T(x, y) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are insulated as shown.

Find the temperature T(x, y) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are insulated as shown.

Find the temperature T(x, y) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are insulated as shown.

These problems should encourage you to experiment with various functions FC:), many of which model interesting flow patterns.(Parallel flow J Show that F(~) = -; K~ (K positive real) describes a uniform flow upward. which can be interpreted as a uniform flow between two parallel lines (parallel planes in three-dimensional space). See Fig. 414. Find the velocity vector, the streamlines. and the equipotential lines.

These problems should encourage you to experiment with various functions FC:), many of which model interesting flow patterns.(Conformal mapping) Obtain the now in Example I from that in Prob. 1 by a suitable conformal mapping.

These problems should encourage you to experiment with various functions FC:), many of which model interesting flow patterns.Find the complex potential of a uniform flow parallel to the x-axis in the positive x-direction.

These problems should encourage you to experiment with various functions FC:), many of which model interesting flow patterns.What happens to the flow in Prob. I if you replace z by ~e-ia with constant 0'. e.g., 0' = 7f/4?

These problems should encourage you to experiment with various functions FC:), many of which model interesting flow patterns.What is the complex potential of an upward parallel flow in the direction of y = 2x?

These problems should encourage you to experiment with various functions FC:), many of which model interesting flow patterns. (Extension of Example l) Sketch or graph the flow in Example I on the whole upper half-plane. Show that you can interpret it as as flow against a horizontal wall (the x-axis)

These problems should encourage you to experiment with various functions FC:), many of which model interesting flow patterns.What F(z) would be suitable in Example I if the angle of the comer were 7f/3?

These problems should encourage you to experiment with various functions FC:), many of which model interesting flow patterns. Sketch or graph the streamlines and equipotential lines of F(:;.) = ;:3. Find V. Find all points at which V l~ parallel to the x-axis.

These problems should encourage you to experiment with various functions FC:), many of which model interesting flow patterns.Find and graph the streamlines of F(:) = :2 + 2:. Interpret the flow.

These problems should encourage you to experiment with various functions FC:), many of which model interesting flow patterns. Show that F(:) = i:2 models a flow around a comer. Sketch the streamlines and equipotential lines. Find V.

These problems should encourage you to experiment with various functions FC:), many of which model interesting flow patterns. (Potential F(z) = lIz) Show that the streamlines of F(~) = l/~ are circles through the origin.

These problems should encourage you to experiment with various functions FC:), many of which model interesting flow patterns.(Cylinder) What happens in Example 2 if you replace ::: by Z2? Sketch and interpret the resulting flow in the first quadrant.

These problems should encourage you to experiment with various functions FC:), many of which model interesting flow patterns.Change F(~) in Example 2 slightly to obtain a flow around a cylinder of radius 1"0 that gives the flow in Example 2 if 1"0 ~ I.

These problems should encourage you to experiment with various functions FC:), many of which model interesting flow patterns.(Aperture) Show that F(:) = arccosh : gives confocal hyperbolas as streamlines, with foci at ::: = ::':: I. and the flow may be interpreted as a flow through an aperture (Fig. 415).

These problems should encourage you to experiment with various functions FC:), many of which model interesting flow patterns.(Elliptical cylinder) Show that F(~) = arccos: gives confocal ellipses as streamlines. with foci at : = ::':: 1. and that the now circulates around an elliptic cylinder or a plate (the segment from -1 to I in Fig. 416).

Verify (3).

Show that every term in (7) is a harmonic function in the disk r < R.

Give the details of the derivation of the series (7) from the Poisson formula (5).

Using (7), find the potential

Using (7), find the potential

Using (7), find the potential

Using (7), find the potential

Using (7), find the potential

Using (7), find the potential

Using (7), find the potential

Using (7), find the potential

Using (7), find the potential

Using (7), find the potential

TEAM PROJECT. Potential in a Disk. (a) Mean value property. Show that the value of a harmonic function

CAS EXPERIMENT. Series (7). Write a program for series developments (7). Experiment on accuracy by computing values from partial sums and comparing them with values that you obtain from your CAS graph. Do (his (a) for Example I and Fig. 421, (b) for

Integrate kl2 around the unit circle. Does your result contradict Theorem I?

VERIFY THEOREM 1 for the given F(::), ::0' and circle of radius l. (::: + 1)3':::0 = 2

VERIFY THEOREM 1 for the given F(::), ::0' and circle of radius l.(::: - 2)2. ::0 = ~

VERIFY THEOREM 1 for the given F(::), ::0' and circle of radius l. 10:::4 '':0 = 0

VERIFY THEOREM 2 for the given cI>(x. y). (xo. Yo) and Circle of radius I. (x - 2)(y - 2), (4. -4)

VERIFY THEOREM 2 for the given cI>(x. y). (xo. Yo) and Circle of radius I.x 2 - .'"2. (3, 8)

VERIFY THEOREM 2 for the given cI>(x. y). (xo. Yo) and Circle of radius I.x 3 - 3xy2, (I, I)

Derive Theorem 2 from Poisson's integral formula.

CAS EXPERIMENT. Graphing Potentials. Graph the potentials in Probs. 5 and 7 and for three other functions of your choice as smfaces over a rectangle or a disk in the xy-plane. Find the locations of maxima and minima by inspecting these graphs.

TEAM PROJECT. Maximum Modulus of Analytic Functions. (a) Verify Theorem 3 for (i) F(:::) = ::2 and the square 4 ~ x ~ 6. 2 ~ Y ~ 4, (ii) F(:::) = e1z and any bounded domain. (iii) F(:::) = sin:: and the unit disk. (b) F(x) = cos x (x real) has a maximum I at O. How does it follow that this cannot be a maximum of IFez.) I = Icos:::1 in a domain containing Z = O? (c) F(::) = 1 + 1::12 is not Lero in the disk Izi ~ 4 and has a minimum at an interior point. Does this contradict Theorem 3? (d) If F(:::) is analytic and not constant in the closed unit disk D: 1<:1 ~ 1 and IF(.;;)I = C = COIlSt on the unit circle. show that F(.:) must have a zero in D. Can you extend this to an arbitrary simple closed curve?

Find the location and si7e of the maximum of IF(;:)I in the unit disk 1:<:1 ;::; IF(;:) = ;:2 - 1

Find the location and si7e of the maximum of IF(;:)I in the unit disk 1:<:1 ;::; IF(;:) = a;: + b la, b complex)

Find the location and si7e of the maximum of IF(;:)I in the unit disk 1:<:1 ;::; IF(;:) = cos 2;:

Verify the maximum principle for <1>(x. y) = eX cos y and the rectangle a ;::; x ;::; b. 0 ;::; y ;::; 2'iT.

Conjugate) Do and a harmonic conjugate \)! of (I) in a region R have their maximum at the same point of R?

(Conformal mapping) Find the location (ul . VI) of the maximum of <1>* = ell cos V in R*: Iwl ;::; 1. V ~ 0, where w = 1I + iv. Find the region R that is mapped onto R* by w = f(;:) = ;:2. Find the potential in R resulting from <1>* and the location (xl' .\'1) of the maximum. Is (lib VI) the image of (Xl' YI)? If so, is this ju~t by chance?

Why can potential problems be modeled and solved by complex analysis? For what dimensions?

What is a harmonic function? A harmonic conjugate?

Give a few example~ of potential problems considered in this chapter.

What is a complex potential? What does it give physically'?

How can conformal mapping be used in connection with the Dirichlet problem?

What heat problems reduce to potential problems? Give a few examples.

Write a short essay on potential theory in fluid flow from memory.

What is a mixed boundary value problem? Where did it occur?

State Poisson's formula and its derivation from Cauchy's formula.

State the maximum modulus theorem and mean value theorems for harmonic functions.

Find the potential and complex potential between the plates y = x and y = x + 10 kept at 10 V and 110 V, respecti vel y.

Find the potential between the cylinders 1:1 = I cm having potential 0 and Id = 10 em having potential 20 kV.

Find the complex potential in Prob. 12.

Find the equipotential line U = 0 V between the cylinders Id = 0.2S cm and 1:1 = 4 cm kept at -220 V and 220 V. respectively. (Guess first.)

Find the potential between the cylinders Izl = 10 cm and 1:::1 = 100 cm kept at the potentials 10 kV and 0, respectively.

Find the potential in the angular region between the plates Arg ::: = 'iT16, kept at 8 k V, and Arg ;: = 'iT13, kept dt 6kV.

Find the equipotential lines of F(;:) = i Ln z..

Find and sketch the equipotential lines of F(:::) = (l + i)/:::.

What is the complex potential in the upper half-plane if the negative half of the x-axis has potential I kV and the positive half is grounded?

Find the potential on the ray)' = x, x > 0, and on the positive half of the x-axis if the positive half of the .v-axis is at 1200 V and the negative half IS grounded.

Interpret Prob. 20 as a problem in heat conduction.

Find the temperature in the upper half-plane if the portion x > 2 of the x-axis is kept at SO"C and the other portion at OC.

Show that the isotherm~ of Fr;:) = _;:::2 + ;: are hyperbolas.

If the region between two concentric cylinders of radii 2 cm and 10 cm contains water and the outer cylinder is kept at 20C, to what temperature must we heat the inner cylinder in order [0 have 30C at distance 5 cm from the axis?

What are the streamlines of Fr:::) = i/;:?

What is the complex potential of a flow around a cylinder of radius 4 without circulation?

Find the complex potential of a source at ::: = 5. What are the streamlines?

Find the temperature in the unit disk 1:1 ;::; I in the form of an infinite series if the left semicircle of Izi = 1 has the temperature of SOC and the right semicircle has the temperature DoC.

Same task as in Prob. 2~ if the upper semicircle is at 40C and the lower at DoC

Find a series for the potential in the unit disk with boundary values <1>( I, 0) = 02 (-'iT < 0 < 'iT).