1 Ivan Avramidi: MATH 332, (Vector Analysis), Final Exam MATH 332: Vector Analysis Sample Final Exams Set 1 1a). Use tensor notation to simplify ( A × B ) · ( C × D ) 1b) Use tensor notation to verify the identity ∇ · ( F × G ) = G · (∇ × F ) − F · (∇ × G ) 2. (a) Sketch the curve x = et cos t, y = et sin t, (0 ≤ t ≤ 1) . z = 0, (b) Determine the arc length of the curve between t = 0 and t = 1. (c) Reparametrize the curve in terms of arc length. 3. Let C be the curve given by the equation R (t) = sin t i + cos t j − ln cos t k 0≤t≤ π 2 Find: (a) the unit tangent T ; (b) the unit normal N ; (c) the curvature k; 4. Let F = xy i + y 2 j + z 2 k . (a) Find the general equation of a flow line. (b) Find the flow line through the point (1, 2, −2). 5. Given the vector field F = (x + xz 2 ) i + xy j + yz k , evalute: (a) div F ; (b) curl F . 6. Let A be a constant vector field, R = x i + y j + z k , and R = | R |. Evaluate 1 ∇× A ×∇ R (Hint: Use tensor notation) 7. Let C be the curve given parametrically by R (t) = 2t i + [t + cos(πt)] j − (t2 + t) k , (0 ≤ t ≤ 1) and F be the vector field F = (eyz + y − z) i + (xzeyz + x) j + (xyeyz − x) k . Compute the line integral R C F · dR. Ivan Avramidi: MATH 332, (Vector Analysis), Final Exam 2 8. Let S be the surface given parametrically by (0 ≤ u ≤ 2, 0 ≤ v ≤ π) , p F = y i − x jRR+ k be a vector RR field and ϕ(x, y, z) = 1 + x2 + y 2 be a scalar field. Compute the surface integrals: (a) S F · dS and (b) S ϕ dS. R = u cos v i + u sin v j + v k , 9. Let S be the portion of the surface of the sphere x2 + y 2 + z 2 = 9 above the plane z = 1 and below the plane z = 2, with normal n pointing upward, and F = (x + yz 3 ) i + xz 3 j − z k be a vector field. Evaluate RR F · n dS. (Hint: This problem can be made reasonably simple if you make use of the divergence theorem.) S 10. Let S be the portion of the surface of the paraboloid z = 9 − x2 − y 2 lying above the plane z = 5, with normal n pointing upward, and F = (x − yz) i + xz j be a vector field. Verify the Stokes’ theorem. 3 Ivan Avramidi: MATH 332, (Vector Analysis), Final Exam Set 2 1a). Use tensor notation to simplify ( A × B ) × ( C × B ) 1b) Use tensor notation to verify the identity ∇ × (∇ × F ) = ∇(∇ · F ) − ∇2 F ) 2. a) Describe and sketch the curve R (t) = t i + sin t j + cos t k . (b) Determine the arc length, L, of the curve between the points (0, 0, 1) and (2π, 0, 1). (c) Reparametrize the curve in terms of arc length s. 3. The position vector of a moving particle is 1 R (t) = (sin t + cot t) i + (sin t − cos t) j + t k . 2 Find: (a) the velocity, v , (b) the speed, v (c) the acceleration, a (d) the unit tangent T , and the unit normal, N , to the path of the particle, in the direction of motion, (e) the curvature, k, of the path. 4. Let F be a vector field defined by F = y i − x j + x k . (a) Find the general (parametric) equation of a flow line. (b) Find the flow line through the point (1, 2, 3). 5. Let R = x i + y j + z k , R = | R |, and F be a vector field defined for R 6= 0 by F = R . R3 Evalute (a) div F and (b) curl F . (Hint: Notice that F = −∇ 1 ). R 6. Let A be a constant vector field, R = x i + y j + z k , and R = | R |. By using the tensor notation show that for R 6= 0 A R ∇ × ( A × ∇R) = + (A · R) . R R H 7. Let F = y i +x j +xyz 2 k . Find the integral C F ·d R around the circumference of the circle x2 −2x+y 2 = 2, z = 1, oriented counterclockwise. (Hint: To parametrize the circle find its center and radius.) 8. Let F be a vector field defined by F = i + xy j , ϕ be a scalar field defined by −1/2 ϕ(x, y, z) = 1 + x2 + 2xy + y 2 − 2z , and S be a surface defined parametrically by R (u, v) = (u + v) i + (u − v) j + u2 k , (0 ≤ u ≤ 1, 0 ≤ v ≤ 1) , RR RR with the normal n pointing upward. Compute the surface integrals: (a) S F · dS and (b) S ϕ dS. (Hint: The integrals over u and v are very simple!) 9. Let F be a vector field defined by F = xy 2 i + yx2 j + z 2 k , and S be the complete surface of the region bounded by the cylinder x2 + y 2 = 4 and by the planes z = 0 and z = 2. Use the divergence theorem to evaluate RR F · n dS. (Hint: Use cylindrical cordinates to evaluate the volume integral.) S 10. Let F be a vector field defined by F = 2y i + (x − 2x3 z) j + xy 3 k , and S be a curved surface of the 2 2 2 hemisphere RR x + y + z = 1, z ≥ 0, with the normal n pointing upward. Use Stokes’ theorem to evaluate the integral S (∇ × F ) · n dS. 4 Ivan Avramidi: MATH 332, (Vector Analysis), Final Exam Set 3 1. Use tensor notation to simplify {[( B × A ) × A ] × A } · C 2. A curve is given by R(t) = et cos t i + et sin t j . Z t dR(t) dR(τ ) ds , (b) find s(t) = = (a) find dτ dτ , (c) find t(s), (d) reparemetrize the curve in terms of dt dt 0 the arc length, s, by substituting t(s) in the equation of the curve, (e) find the values of t corresponding to the points P1 (1, 0, 0) and P2 (−eπ , 0, 0), (f) find the length of the curve, L, between the points P1 and P2 by using the arc length function s(t). 3. A particle moves so that its position R at time t is given by R(t) = e−t cos t i + e−t sin t j + e−t k Find: (a) the velocity, v , (b) the speed, | v |, (c) the unit tangent, T , (d) the unit normal, N , and (e) the curvature, k, of the path. 4. Let F be a vector field defined by F = x2 i + y j + k . Find the general (parametric) equation of a flow y line. 5. Let F be a vector field defined by F = x2 y i + z j − (x + y − z) k . Find: (a) div F , (b) curl F , and (c) grad div F . 6. Let A be a constant vector field and ϕ be a scalar field defined by ϕ = 1 + x + y + z + xyz. By using the tensor notation show that curl ( A × grad ϕ) + grad ( A · grad ϕ) = 0 . (Hint: This equation holds for any harmonic scalar field, i.e. a scalar field satisfying the Laplace equation ∆ϕ = 0, so the particular form of ϕ is not important!) 7. Let F = (3x + 4y) i + (2x + 3y 2 ) j + (exyz ) kI and C be the counterclockwise oriented circle of radius 2 in F · d R around C. (Hint: The equations of the circle are: the xy plane with the center at the origin. Find 2 C 2 x + y = 4, z = 0. Parametrize the circle by R (t) = 2 cos t i + 2 sin t j , 0 ≤ t ≤ 2π.) 2 8. Let F be a vector field defined by F = x i + y j + (zRR − 1) k and S be a closed surface bounded by the 2 2 planes z = 0, z = 1 and the cylinder x + y = 1. Find S F · d S . (Hint: The surface S consists of three parts. Find the normals to each part and compute the surface integrals over each part separately.) 9. Let F be a vector field defined by F = x i − y j , and D be a domain in space bounded by the planes z = 0, z = 1 and the cylinder x2 + y 2 = 1. Verify the Divergence Theorem, i.e. compute both sides of the divergence theorem and show that they are equal. (Hint: The boundary of D consists of three parts. Find the normals to each part compute the surface integrals over each part separately. Use cylindrical coordinates for the volume integral.) 10. 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 − x2 − y 2 that Zlies Z above the plne z = 0 (with the normal n pointing upward). Use Stokes’ curl F · n dS. theorem to evaluate the integral S