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mathematics
calculus

Questions and Answers of Calculus

A buffalo (Figure 29) stampede is described by a velocity vector field F =xy ˆ’ y3, x2 + ykm/h in the region D defined by 2 ‰¤ x ‰¤ 3, 2 ‰¤ y
Let n be the outward pointing unit normal vector to a simple closed curve C. The normal derivative of a function (, denoted (( / (n, is the directional derivative Dn (() = (( ( n. Prove thatWhere D
Prove that m(r) ‰¤ IÏ•(r) ‰¤ M(r), where m(r) and M(r) are the minimum and maximum values of Ï• on Cr . Then use the continuity of ( to prove that
The line integral of F = (ex+y, ex-y) along the curve (oriented clockwise) consisting of the line segments by joining the points (0, 0), (2, 2), (4, 2), (2, 0), and back to (0, 0) (note the
LetAnd let C be the quarter-circle path from A to B in Figure 17. Evaluate As follows: (b) Show that the line integrals of G along the segments OÌ…AÌ… and
EvaluateFor the non-closed path ABCD in Figure 19. Use the method of Exercise 12.
In Exercises 15 - 18, use Eq. (6) to calculate the area of the given region. The circle of radius 3 centered at the origin?
Indicate with an arrow the boundary orientation of the boundary curves of the surfaces in Figure 14, oriented by the outward-pointing normal vectors?
Which condition on F guarantees that the flux through S1 is equal to the flux through S2 for any two oriented surfaces S1 and S2 with the same oriented boundary?
In Exercises 1-2, calculate curl(F).1.2.
Let S be the surface of the cylinder (not including the top and bottom) of radius 2 for 1 ‰¤ z ‰¤ 6, oriented with outward-pointing normal (Figure 16).(a) Indicate with an arrow the orientation
Let A be the vector potential and B the magnetic field of the infinite solenoid of radius R in Example 6. Use Stokes' Theorem to compute: (a) The flux of B through a circle in the xy-plane of radius
A uniform magnetic field B has constant strength b in the z-direction [that is, B = (0, 0, b)].(a) Verify that A = 12B Ã— r is a vector potential for B, where r = (x, y, 0).(b) Calculate
LetUse Stokes' Theorem to find a plane with equation ax + by + cz = 0 (where a, b, c are not all zero) such that for every closed C lying in the plane. Choose a, b, c so that curl(F) lies in the
For any two closed curves lying on a cylinder whose central axis is the z-axis (Figure 21).
You know two things about a vector field F:(i) F has a vector potential A (but A is unknown).(ii) The circulation of A around the unit circle (oriented counterclockwise) is 25.Determine the flux of F
Prove that curl(fa) = ∇f × a, where f is a differentiable function and a is a constant vector?
Prove the following Product Rule:curl(f F) = f curl(F)+∇f × F
Verify that B = curl(A) for r > R in the setting of Example 6?
You know two things about a vector field F:(i) F has a vector potential A (but A is unknown).(ii) The circulation of A around the unit circle (oriented counterclockwise) is 25.Determine the flux of F
What is the flux of F = (1, 0, 0) through a closed surface?
Justify the following statement: The flux ofthrough every closed surface is positive.
Which of the following statements is correct (where F is a continuously differentiable vector field defined everywhere)?(a) The flux of curl(F) through all surfaces is zero.(b) If F = ∇ϕ, then the
How does the Divergence Theorem imply that the flux of F = {x2, y − ez, y − 2zx} through a closed surface is equal to the enclosed volume?
In Exercises 1-2, compute the divergence of the vector field. 1. F = {xy, yz, y2 - x3} 2. F = {x - 2zx2, z - xy, z2x2}
F = (0, 0, z3 / 3). S is the sphere x2 + y2 + z2 = 1?
F = (x3, 0, z3), S is the octant of the sphere x2 + y2 + z2 = 4, in the first octant x ( 0, y ( 0, z ( 0?
F = {x + y, z, z - x}, S is the boundary of the region between the paraboloid z = 9 - x2 - y2 and xy-plane?
Let S be the half-cylinder x2 +y2 = 1, x ‰¥ 0, 0 ‰¤ z ‰¤ 1.Assume that F is a horizontal vector field (the z-component is zero) such that F(0, y, z) = zy2i. Let W be
Use Eq. (10) to calculate the volume of the unit ball as a surface integral over the unit sphere?
Let W be the region between the sphere of radius 4 and the cube of side 1, both centered at the origin. What is the flux through the boundary S = ∂W of a vector field F whose divergence has the
Find and prove a Product Rule expressing div(fF) in terms of div(F) and (f?
Prove that div (∇ f ×∇g) = 0?
Show that F = {2yz -xy, y, yz - z} has a vector potential and find one?
Find a constant c for which the velocity fieldOf a fluid is incompressible [meaning that div(v) = 0]?
Use Green's Theorem to evaluate the line integral around the given closed curve.Where C is the rectangle ˆ’1 ‰¤ x ‰¤ 2, ˆ’2 ‰¤ y ‰¤
Calculate the curl and divergence of the vector field. F = ((e-x2-y2-z2)?
Recall that if F1, F2, and F3 are differentiable functions of one variable, thenUse this to calculate the curl of
Prove that if F is a gradient vector field, then the flux of curl(F) through a smooth surface S (whether closed or not) is equal to zero?
Letand let S be the upper half of the ellipsoid Oriented by outward pointing normals. Use Stokes Theorem to compute ( (s curl (F) ( dS?
Let S be the side of the cylinder x2 + y2 = 4, 0 ≤ z ≤ 2 (not including the top and bottom of the cylinder). Use Stokes' Theorem to compute the flux of F = (0, y,−z) through S (with outward
Use the Divergence Theorem to calculate ( (s F ( dS for the given vector field and surface. F = {xy, zy, x2z + z2}, S is the boundary of the box [0, 1] ( [2, 4] ( [1, 5]?
F = {xyz + xy, ½ y2 (1 - z) + ex,ex2 + y2}, S is the boundary of the solid bounded by the cylinder x2 + y2 = 16 and the planes z = 0 and z = y - 4?
Use Green's Theorem to evaluate the line integral around the given closed curve.Where C consists of the arcs y = x2 and y = (x, 0 ( x ( 1, oriented clockwise?
Calculate the curl and divergence of the vector field. F =yi - zk
Verify that(a) (ˆš2 - i) - i(1 - ˆš2i) = -2i;(b) (2, - 3)(-2, 1) = (-1, 8)(c)
Solve the equation z2 + z + 1 = 0 for z = (x, y) by writing (x, y) (x, y) + (x, y) + (1, 0) = (0, 0) and then solving a pair of simultaneous equations in x and y. Suggestion: Use the fact that no
Show that (a) Re(iz) = − Im z; (b) Im(iz) = Re z.
Show that (1 + z)2 = 1 + 2z + z2.
Prove that multiplication of complex numbers is commutative, as stated at the beginning of Sec. 2.
Verify (a) The associative law for addition of complex numbers, stated at the beginning of Sec. 2; (b) The distributive law (3), Sec. 2.
Reduce each of these quantities to a real number:(a)(b) (c) (1 - i)4.
With the aid of relations (10) and (11) in Sec. 3, derive the identity
Use the identity obtained in Exercise 6 to derive the cancellation law
Locate the numbers z1 + z2 and z1 − z2 vectorially when (a) z1 = 2i, z2 = 2/3 - I; (b) z1 = (-√3, 1), z2 = (√3,0); (c) z1 = (-3, 1) z2 = (1, 4); (d) z1 = x1 + iy1, z2 = x1 - iy1.
Verify inequalities (3), Sec. 4, involving Re z, Imz, and |z|. |z1 + z2| ≤ |z1| + |z2|.
Verify that √2 | z | ≥ |Re z| + | Im z |.
In each case, sketch the set of points determined by the given condition: |z − 1 + i| = 1;
Using the fact that |z1 − z2| is the distance between two points z1 and z2, give a geometric argument that (a) |z − 4i| + |z + 4i| = 10 represents an ellipse whose foci are (0, ± 4); (b) |z −
Use properties of conjugates and moduli established in Sec. 5 to show that
Prove that (a) z is real if and only if = z; (b) z is either real or pure imaginary if and only if 2 = z2.
Use mathematical induction to show that when n = 2, 3, . . . ,
Show that the equation |z − z0| = R of a circle, centered at z0 with radius R, can be written |z|2 − 2Re(z0) + |z0|2 = R2.
Using expressions (6), Sec. 5, for Re z and Imz, show that the hyperbola x2 − y2 = 1 can be written z2 + 2 = 2.
Sketch the set of points determined by the condition (a) Re( − i) = 2; (b) |2 + i| = 4.
Verify properties (3) and (4) of conjugates in Sec. 5.(3)(4)
Use property (4) of conjugates in Sec. 5 to show that (a) z1z2z3 = z1 z2 z3; (b) 4 = 4.
Use results in Sec. 5 to show that when z2 and z3 are nonzero,(a)(b)
Show that |Re(2 + + z3)| ≤ 4 when | z | ≤ 1.
By factoring z4 ˆ’ 4z2 + 3 into two quadratic factors and using inequality (8), Sec. 4, show that if z lies on the circle |z| = 2, then
Find the principal argument Arg z when (a) z = i/-2 - 2i; (b) z = (√3 - i)6.
Use de Moivre's formula (Sec. 7) to derive the following trigonometric identities: (a) cos 3θ = cos3 θ − 3 cos θ sin2 θ; (b) sin 3θ = 3 cos2 θ sin θ − sin3 θ.
Using the fact that the modulus |eiθ − 1| is the distance between the points eiθ and 1 (see Sec. 4), give a geometric argument to find a value of θ in the interval 0 ≤ θ < 2π that satisfies
Let z be a nonzero complex number and n a negative integer (n = ˆ’1,ˆ’2, . . .). Also, write z = reiÎ¸ and m = ˆ’n = 1, 2, . . . . Using the
Prove that two nonzero complex numbers z1 and z2 have the same moduli if and only if there are complex numbers c1 and c2 such that z1 = c1c2 and z2 = c1c2.and [see Exercise 2(b)]
Establish the identityand then use it to derive Lagrange's trigonometric identity:
Find the square roots of (a) 2i; (b) 1 − √3i and express them in rectangular coordinates.
In each case, find all the roots in rectangular coordinates, exhibit them as vertices of certain squares, and point out which is the principal root: (a) (−16)1/4; (b) (−8 - 8 √3i)1/4.
In each case, find all the roots in rectangular coordinates, exhibit them as vertices of certain regular polygons, and identify the principal root: (a) (−1)1/3; (b) 81/6.
According to Sec. 9, the three cube roots of a nonzero complex number z0 can be written c0, c0Ï‰3, c0Ï‰23 where c0 is the principal cube root of z0 andShow that if z0 =
(a) Let a denote any fixed real number and show that the two square roots of a + i Arewhere A = ˆša2 + 1 and Î± = Arg(a + i). (b) With the aid of the trigonometric identities
Find the four zeros of the polynomial z4 + 4, one of them being z0 = √2 eiπ/4 = 1 + i. Then use those zeros to factor z2 + 4 into quadratic factors with real coefficients.
Show that if c is any nth root of unity other than unity itself, then 1 + c + c2 +· · ·+cn−1 = 0.
Let z = reiθ be a nonzero complex number and n a negative integer (n = −1, −2, . . .). Then define z1/n by means of the equation z1/n = (z−1)1/m where m = −n. By showing that the m values of
Sketch the following sets and determine which are domains: (a) |z − 2 + i| ≤ 1; (b) |2z + 3| > 4; (c) Im z > 1; (d) Im z = 1; (e) 0 ≤ arg z ≤ π/4 (z ≠ 0); (f) |z − 4| ≥ |z|.
In each case, sketch the closure of the set:(a) ˆ’Ï€ = 0);(b) |Re z| (c)(d) Re(z2) > 0.
Let S be the open set consisting of all points z such that |z| < 1 or |z − 2| < 1. State why S is not connected.
Prove that if a set contains each of its accumulation points, then it must be a closed set.
For each of the functions below, describe the domain of definition that is understood: (a) f(z) = 1/z2 + 1; (b) f(z) = Arg(1/z); (c) f(Z) = z/ z+ 1; (d) f(z) = 1/1 - |z|2.
Suppose that f (z) = x2 ˆ’ y2 ˆ’ 2y + i(2x ˆ’ 2xy), where z = x + iy. Use the expressions (see Sec. 5)to write f(z) in terms of z, and simplify the result.
Use the theorem in Sec. 17 to show that(a)(b) (c)
With the aid of the theorem in Sec. 17, show that when(a) (b)
Show that the limit of the functionas z tends to 0 does not exist. Do this by letting nonzero points z = (x, 0) and z = (x, x) approach the origin.
Use results in Sec. 20 to find f'(z) when(a) f (z) = 3z2 ˆ’ 2z + 4;(b) f (z) = (1 ˆ’ 4z2)3;(c)(d)
Apply definition (3), Sec. 19, of derivative to give a direct proof that
Suppose that f(z0) = g(z0) = 0 and that f'(z0) and g'(z0) exist, where g'(z0) ‰ 0. Use definition (1), Sec. 19, of derivative to show that
(a) Recall (Sec. 5) that if z = x + iy, thenBy formally applying the chain rule in calculus to a function F(x, y) of two real variables, derive the expression (b) Define the operator suggested by
From results obtained in Secs. 21 and 22, determine where f'(z) exists and find its value when (a) f (z) = 1/z; (b) f (z) = x2 + iy2; (c) f (z) = z Im z.
Use the theorem in Sec. 23 to show that each of these functions is differentiable in the indicated domain of definition, and also to find f'(z): (a) f (z) = 1/z4 (z ≠ 0); (b) f (z) = √reiθ/2 (r
Show that when f(z) = x3 + i(1 − y)3, it is legitimate to write f'(z) = ux + ivx = 3x2 only when z = i.

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