MATH 332: Vector Analysis Summer 2005 Homework

Transcription

1 MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9, 1.10, 1.12) 1. Prove that for any two vectors A and B there holds the triangle inequality 2. Let D be a vector defined by A + B A + B. D = 10 i 10 j + 5 k and P be a point with Cartesian coordinates (1, 1, 3). Find an equation of the plane orthogonal to the vector D and passing through the point P. 3. Let A, B, and C be three points in space and A, B and C be the corresponding vectors pointing from the origin to these points. Assume that the points A, B and C do not lie on the same line, that is, the vectors A B and B C are not parallel. Let D be the vector D = A B + B C + C A. Show that the vector D is orthogonal to the plane passing through the points A, B and C. Hint: Consider ( B A ) ( C A ). 4. Let u and v be two unit vectors orthogonal to each other. Let w = u v. Show that: a) w is a unit vector orthogonal to both u and v, and b) v = w u and u = v w. 5. Let B be an arbitrary vector with B < 1. Let A be an arbitrary vector and { A n } n=1 be a sequence of vectors defined by A 1 = A, A n+1 = B A n. What is the ultimate behavior of this sequence? What can you say about the direction and the length of the vectors A n as n?

2 MATH 332, (Vector Analysis), Summer 2005: Homework 2 Set 2. (Triple Scalar Product, Vector Identities, Tensor Notation) Sections: 1.13, 1.14, Let { u, v, w } be a right-handed orthonormal system. Let A be a vector defined by A = i u + j v + k w. Show that the angle between the vectors A and i is equal to the angle between the vectors A and u. 2. Let { A, B, C } be an arbitrary system of noncoplanar vectors. Show that any vector V can be decomposed in terms of the vectors A, B and C as V = [ V, B, C ] [ A, B, C ] A + [ V, C, A ] [ A, B, C ] B + [ V, A, B ] [ A, B, C ] C. 3. Use tensor methods to simplify the expressions 4. Use tensor methods to show that 5. Use tensor methods to simplify 1) ( A B ) ( C D ) 2) A B ) ( C D ) ( ) 3) [ A ( A B )] A C 4) ( A B ) (( B C ) ( C A ) ). A ( B C ) + B ( C A ) + C ( A B ) = 0. A B 2 + ( A B ) 2 A 2 B 2 6. The determinant of a 2 2 matrix is defined by ( ) a b det = ad bc. c d Use tensor methods to express ( A B ) ( A B ) as a determinant of a 2 2 matrix involving only scalar products.

3 MATH 332, (Vector Analysis), Summer 2005: Homework 3 Set 3. (Differentiation, Space Curves, Velocity, Speed, Arc Length, Tangent) Sections: 2.1, Let F (t) be a vector-valued function defined by F (t) = sin t i + cos t j + k. a) Find F (t). b) Show that F (t) is parallel to the xy-plane for any t. c) For what values of t is F (t) parallel to the xz-plane? d) Does F (t) have constant length? e) Does F (t) have constant length? f) Find F (t). 2. Show that for any vector-valued function F, ( d F d F ) = F d2 F dt dt dt Let R (t) be a curve defined by R (t) = (sin t t cos t) i + (cos t + t sin t) j + t 2 k. Find: a) the arc length between the points (0, 1, 0) and ( 2π, 1, 4π 2 ); b) the unit tangent T (t); c) T (π). 4. Let R (t) be a curve defined by R (t) = t i + sin t j + cos t k. 2π a) What curve is this geometrically? b) Find the arc length between the points (0, 0, 1) and (1, 0, 1). c) Find the unit tangent T (t). d) Compute the unit tangent T at the point (0, 0, 1). 5. Show that the curve intersects the plane at right angles. R (t) = t i + 2t 2 j + t 3 k. x + 8y + 12z = Parametrize a right-handed helix with unit pitch that is wrapped around the cylinder y 2 + z 2 = 1.

4 MATH 332, (Vector Analysis), Summer 2005: Homework 4 Set 4. (Acceleration, Curvature, Normal, Binormal, Torsion, Frenet Formulas) Sections: Let the position vector of a moving particle be given by R (t) = e t cos t i + e t sin t j + e t k. Find: a) the speed of the particle, b) the tangential and the normal components of acceleration, c) the unit tangent to the curve, and d) the curvature of the curve. 2. Let a curve be described by R (t) = sin t i + cos t j ln cos t k. Find: a) the element of arc length ds; b) the unit tangent T ; c) the principal normal N ; and d) the curvature. 3. Let F (t) be a smooth vector-valued function. Find {( d F d F ) d2 F dt dt dt 2 4. Let R (s) be a curve in the natural parametrization such that R (s) = C, where C is a constant (that is, the curve lies on the sphere). Show that R = ρ N 1 τ dρ ds B, where ρ is the radius of curvature, N is the principal normal, τ is the torsion and B is the binormal. Hint: Keep differentiating R R = C, using the Frenet formulas. 5. Let R (s) be a curve in natural parametrization and T, N and B be the unit tangent, principal normal and the binormal, and κ and τ are the curvature and the torsion of the curve. The Darboux vector ω is defined by ω = τ T + κ B. Show that the following equations are satisfied }. d T ds = ω T d N ds = ω N d B ds = ω B.

5 MATH 332, (Vector Analysis), Summer 2005: Homework 5 Set 5. (Scalar and Vector Fields, Level Surfaces, Directional Derivative, Gradient, Flow Lines) Sections: 3.1, Let f be a scalar field defined by f(x, y, z) = R = x 2 + y 2 + z 2. a) Find the directional derivative of f at the point P with coordinates (1, 3, 2) in the direction of the vector A = i + 2 j + 2 k. b) Find the direction of maximal increase of f at P. c) Find the magnitude of the greatest rate of change of f at P. 2. Find an equation of the plane tangent to the surface at the point (2, 3, 13). 3. Let ϕ be a scalar field defined by z = x 2 + y 2 ϕ(x, y, z) = x 2 y + zy + z 3. Find: a) grad ϕ; b) the equation of the tangent plane to the level surface of ϕ at the point (1, 1, 1). 4. Let f be a scalar field, f(x, y, z) = C be its level surface, and R (t) be a curve that lies on this level surface. Show that What does this mean geometrically? d R dt grad f = Let f and g be scalar fields and h = fg. Show that Hint: Use tensor methods. 6. Let F be a vector field defined by grad h ( grad f grad g) = 0. F = x 2 i + y 2 j + k. Find: a) the general equation of a flow line; b) the flow line passing through the point (1, 1, 2).

6 MATH 332, (Vector Analysis), Summer 2005: Homework 6 Set 6. Divergence, Curl, Laplacian, Tensor Notation) Sections: 3.3, 3.4, 3.5, Let R = x i + y j + z k be the position vector field, R = R = x 2 + y 2 + z 2 and F be a vector field defined by Find div F. (Use tensor methods.) F = R R Let R = x i + y j + z k, R = R = x 2 + y 2 + z 2, and ϕ be a scalar field defined by ϕ = 1 R. Find: a) F = grad ϕ; b) ϕ = div F = div grad ϕ. (Use tensor methods.) 3. Let F be a vector field defined by Find: a) div F, b) curl F. 4. Let R = x i + y j + z k. Compute F = (x + xz 2 ) i + xy j + yz k. div R, and curl R. 5. Let R = x i + y j + z k, R = R = x 2 + y 2 + z 2, and F be a vector field defined by F = f(r) R, where f(r) is an arbitrary smooth function. Find curl F. (Use tensor methods.)

7 MATH 332, (Vector Analysis), Summer 2005: Homework 7 Set 7. (Differential Vector Identities, Tensor Notation) Sections: 3.8, Let R = x i +y j +z k, R = R, and A be a constant vector field. By using the tensor methods show that ( ) A R div = 0. R 2. Let R = x i + y j + z k, R = R, A and B be constant vector fields, and F be a vector field defined by F = A f( R B ), where f is a differentiable function of a single variable. Show that curl F is orthogonal to both A and B. (Use tensor methods.) 3. Let R = x i + y j + z k, R = R and A be a constant vector field. Use tensor methods to evaluate a) curl (R 2 A ), b) div ( A R ), c) curl ( A R ), d) grad ( A R ). 4. Let R = x i + y j + z k, R = R and A be a constant vector field. Use tensor methods to evaluate ( grad A grad 1 ) ( + curl A grad 1 ). R R

8 MATH 332, (Vector Analysis), Summer 2005: Homework 8 Set 8. (Cylindrical and Spherical Coordinates, Curvilinear Coordinates) Sections: 3.10, Let f be a radial scalar field that depends on r only (in spherical coordinates). Show that f = f (r) + 2 r f (r). 2. Let F and G be vector fields defined by F = x i + y j x 2 + y 2, G = y i + x j x 2 + y 2. Find the divergence and the curl of F and G in cylindrical coordinates. 3. Find the arc length of the curve described in spherical coordinates by where 0 t π. r = sin t, ϕ = t, θ = π 2, 4. Let F be a vector field defined in spherical coordinates by Compute div F and curl F. F = e r + r e ϕ + r cos ϕ e θ. 5. Parabolic cylindrical coordinates (u, v, z) are defined by x = 1 2 (u2 v 2 ), y = uv, z = z, where u, v 0, and z. a) Determine the scale factors h u, h v, and h z. b) Find the arc length ds and the volume element dv ; c) Find the formula for divergence.

9 MATH 332, (Vector Analysis), Summer 2005: Homework 9 Set 9. (Line Integrals, Conservative Vector Fields, Scalar Potential) Sections: 4.1, Let F be a vector field defined by F = y i + x j + xyz 2 k and C be the circle x 2 2x + y 2 = 2 in the plane z = 1 oriented counterclockwise. Find the circulation F d R of the vector field F around C. C 2. Let F be a vector field defined by F = [y + yz cos(xyz)] i + [x 2 + xz cos(xyz)] j + [z + xy cos(xyz)] k, and C be the ellipse parametrized by R (t) = 2 cos t i + 3 sin t j + k, where 0 t 2π. Evaluate the circulation C F d R of the vector field F around C. 3. Let F be a vector field defined by F = (sin x + y 2 ) i + (x e y ) j, and C be the boundary of the semicircular region x 2 + y 2 4, y 0, oriented counterclockwise. Evaluate C F d R. 4. Let F be a vector field defined by F = (x + y) i + (y + z) j + (z + x) k, and C be the curve of intersection of the plane x + y + z = 1 with the cylinder x 2 + y 2 = 1 oriented counterclockwise as viewed from above. Compute C F d R. 5. Let F be a vector field such that it is everywhere parallel to R. Let C be a curve on a sphere centered at the origin. Show that F d R = Show that the following vector fields are not conservative C a) F = y i + x j, b) F = x i + y j x 2 + y 2,

10 MATH 332, (Vector Analysis), Summer 2005: Homework Let F be a vector field defined by F = y i + x j x 2 + y 2. and C be the circle of radius r centered at the origin in the xy-plane oriented counterclockwise. Compute C F d R. 8. Show that the vector field is conservative. 9. Show that the vector field is conservative and find its scalar potential. F = 2xy i + (x 2 + z) j + y k F = [y + z cos(xz)] i + x j + x cos(xz) k

11 MATH 332, (Vector Analysis), Summer 2005: Homework 11 Set 10. (Irrotational Vector Fields, Solenoidal Vector Fields, Vector Potential) Sections: 4.4, Use the zero curl test to determine whether the following vector fields are conservative: a) F = sin x i + y 2 j + e z k, b) F = 2x x 2 + y i + 2y 2 x 2 + y j + 2z k. 2 If the zero curl test is not applicable, try to find the the scalar potential to determine whether a field is conservative. 2. Let R = x i + y j + z k, R = R, and F be a vector field defined by F = R R 3. Show that the vector field F is conservative and the scalar field is a scalar potential for the vector field F. 3. Let F be a vector field defined by and C be curve a parametrized by where 0 t 1. Evaluate C F d R. 4. Show that the vector field ϕ = 1 R F = (6x 2e 2x y 2 ) i 2ye 2x j + cos z k, R (t) = t i + (t 1)(t 2) j + π 2 t3 k, F = (1 + x)e x+y i + [xe x+y + 2y] j 2z k is conservative by finding the scalar potential for it. Let G be a vector field defined by and C be a curve parametrized by G = (1 + x)e x+y i + [xe x+y + 2z] j 2y k R (t) = (1 t)e t i + t j + 2t k, where 0 t 1. Use the similarity between the vector fields F and G to evaluate C G d R.

12 MATH 332, (Vector Analysis), Summer 2005: Homework Show that the vector field is solenoidal and find its vector potential. F = x j 6. Let R = x i + y j + z k, R = R, A be a constant vector field, and F be a vector field defined by F = A R R 2. By using tensor methods show that the vector field is a vector potential for the vector field F. G = ( A R ) R R 2, 7. Let F and G be irrotational vector fields and V be a vector field defined by Show that V is solenoidal. 8. Let F be a vector field defined by V = F G. F = (x 2 y 2 ) i 2xy j. Show that F is conservative and solenoidal at the same time and find the scalar potential and vector potential for F.

13 MATH 332, (Vector Analysis), Summer 2005: Homework 13 Set 11. (Surfaces, Surface Integrals) Sections: 4.6, Let S be a surface parametrized by R (u, v) = u 2 i + 2uv j + v 2 k. Find the surface elements ds and d S in terms of du and dv. 2. Let S be a surface parametrized by R (u, v) = (1 + cos u) cos v i + (1 + cos u) sin v j + sin v k. Find the surface elements ds and d S in terms of du and dv. 3. Let S be a right circular cylinder parametrized by R (u, v) = a cos u i + a sin u j + v k. Find the surface elements ds in terms of du and dv. 4. Let S be a paraboloid of revolution parametrized by R (u, v) = u i + v j + (u 2 + v 2 ) k. Find the surface elements ds in terms of du and dv. 5. Find the area of the surface S parametrized by where 0 u 1 and 0 v Let F and G be vector fields defined by R (u, v) = u 2 i + uv j v2 k, F = x i y j, G = x i + y j + (z 2 1) k, and S be a closed surface bounded by the planes z = 0, z = 1 and the cylinder x 2 +y 2 = a 2 oriented by the unit outward normal n. Evaluate the fluxes S F n ds and S G n ds of the vector fields F and G through the surface S. 7. Let F be a vector field defined by and S be a surface parametrized by F = y i x j + xy k, R (u, v) = u 2 i + uv j v2 k, where 0 u 1 and 0 v 3. Evaluate S F d S.

14 MATH 332, (Vector Analysis), Summer 2005: Homework Let F be a vector field defined by F = y i + (x + 2) j + x 3 sin(yz) k, and S be a portion of the cylinder x 2 + y 2 = 1 that lies in the first octant and below z = 1. Evaluate S F d S.

15 MATH 332, (Vector Analysis), Summer 2005: Homework 15 Set 12. (Volume Integrals, Divergence Theorem) Sections: 4.8, 4.9, Let D be the region bounded by the sphere S = D of radius a centered at the origin oriented by the outward unit normal n, and let R = x i + y j + z k. Evaluate the integrals a) div R dv b) R d S. 2. Let D be the region bounded by the surface D z = e (x2 +y 2), the cylinder x 2 + y 2 = 1 and the plane z = 0. Find the volume of the region D. 3. Let F and G be vector fields defined by F = x i y j, G = x i + y j + (z 2 1) k, and S be a closed surface bounded by the planes z = 0, z = 1 and the cylinder x 2 +y 2 = a 2 oriented by the unit outward normal n. Use the divergence theorem to evaluate the fluxes F n ds and G n ds of the vector fields F and G through the surface S. S S 4. Let F be a vector field defined by F = y 2 x i + x 2 y j + z 2 k, and S = D be the complete surface of the region D bounded by the cylinder x 2 + y 2 = 4 and by the planes z = 0 and z = 2 oriented by the outward unit normal n. Use the divergence theorem to evaluate the flux S F n ds. 5. Let R = x i +y j +z k, R = R, and S be a surface of a sphere of radius a and centered at the origin. Use the divergence theorem to find the flux S R R n ds. 6. Let F is a vector such that div F = 0 everywhere except at the origin. Let D be the region described by 1 x 2 + y 2 + z 2 4. a) Find the boundary S = D of the region D oriented by the outward unit normal n. b) Compute the flux F n ds. c) The boundary S = D of the region D consists of S two disconnected parts S 1 and S 2. Show that the fluxes of the vector field F through S 1 and S 2 satisfy S 1 F n ds = S 2 F n ds. S

16 MATH 332, (Vector Analysis), Summer 2005: Homework Let R = x i + y j + z k, R = R, and F be a vector field defined by F = R R. 3 a) Show that div F = 0 everywhere except at the origin. b) Find the flux F n ds S of the vector field F through the surface S of the unit sphere centered at the origin. c) Find the flux F n ds of the vector field F through the surface S of the unit sphere S centered at the point (4, 0, 0).

17 MATH 332, (Vector Analysis), Summer 2005: Homework 17 Set 13. (Green Theorem, Stokes Theorem) Sections: 4.9, 5.4, Let F be a vector field defined by F = (3x + 4y) i + (2x + 3y 2 ) j, and C be a circle x 2 + y 2 = 4 oriented counterclockwise. Use Stokes theorem to evaluate the circulation C F d R. 2. Let F be a vector field defined by F = y i + x j + xyz 2 k, and C be a circle x 2 2x + y 2 = 2 in the plane z = 1 oriented counterclockwise viewed from above. Use Stokes theorem to evaluate the circulation C F d R. 3. Let F be a vector field defined by F = x i + (x + y) j + (x + y + z) k, and C be an ellipse x 2 + y 2 = 1, z = y, oriented counterclockwise viewed from above. Use Stokes theorem to evaluate the circulation C F d R. 4. Let F be a vector field defined by F = y i + (x 2x 3 z) j + xy 3 k, and S be the surface of a sphere x 2 + y 2 + z 2 = a 2 above the xy-plane oriented by an upward normal n. Use Stokes theorem to evaluate S curl F d S. 5. Let F be a vector field defined by F = (y z) i (x + z) j + (x + y) k, and S be the portion of the paraboloid z = 9 x 2 y 2 that lies above the plane z = 0 and oriented by an upward normal n. Use Stokes theorem to evaluate S curl F d S. 6. Let F be a vector field defined by F = 2y i + (x 2x 3 z) j + xy 3 k, and S be the curved surface of the semisphere x 2 + y 2 + z 2 = 1, z 0, oriented by an upward normal n. Use Stokes theorem to evaluate S curl F d S.

18 MATH 332, (Vector Analysis), Summer 2005: Homework Let F be a vector field defined by and C be the curve parametrized by F = ye x i + (x + e x ) j + z 2 k, R (t) = (1 + cos t) i + (1 + sin t) j + (1 sin t cos t) k, where 0 t 2π. Use Stokes theorem to evaluate F d R. Hint: Observe that C is C contained in a certain plane and that the projection of C on the xy-plane is a circle. 8. Let F be a vector field defined by F = 3y i + (5 2x) j + (z 2 2) k, and S be the open hemisphere x 2 +y 2 +z 2 = 4 above the xy-plane oriented by the upward normal n. Find curl F n ds. S 9. Let F be a vector field such that curl F = 2y i 2z j + 3 k. Let S be the sphere x 2 + y 2 + z 2 = 9 oriented by an outward normal and U be the upper (open) hemisphere x 2 + y 2 + z 2 = 9, z > 0, oriented by the upward normal. Evaluate curl F n ds and curl F n ds. S U