PRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives.

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1 PRACTICE PROBLEMS Please let me know if you find any mistakes in the text so that i can fix them Let Show that f is C 1 and yet How is that possible? 1. Mixed partial derivatives f(x, y) = {xy x y x +y, (x, y) (0, 0) 0, (x, y) = (0, 0) f xy (0, 0) f yx (0, 0). Double Integrals.1. Compute the volume of the sphere of radius R by any method you wish... Compute e x dx. Hint: evaluate the double integral e x y dxdy in two different ways. R.3. Compute x y dxdy where D is the triangle with vertices (0, 0), (1, 0), (1, ). D.4. Set up a double integral in polar coordinates to find the volume of the solid that is bounded by the paraboloid z = x + y and the plane x = z. Do not evaluate the integral..5. Evaluate the integral y cos( πx ) dx dy.6. Consider the iterated integral ( 4 I = x y e x y a) Sketch the region of integration. b) Reverse the order of integration. c) Evaluate I. 3 ) dx dy.7. Let D the bounded domain in R enclosed by the parabola y = x and the line y = x. a) Parametrize this domain using polar coordinates. b) Use the polar parametrization to evaluate the integral dxdy x + y D 1

2 PRACTICE PROBLEMS.8. a) Compute D x ydxdy where D is the domain in R bounded by the lines y = 0, y = 1 x, and y = x + 1. b) Use Green s theorem to compute the integral from part a)..9. a) Compute [ 1,1] [0,1] (x + y )dxdy. b) Use Green s theorem to compute the integral from part a)..10. Compute the volume of the region in R 3 bounded by the paraboloid z = 4 x y and the xy-plane. 3. Triple integrals 3.1. The shape of a platform is given by x + y ( z), 0 z 1. a) Describe the shape in cylindrical coordinates. b) Compute the volume of the platform. 3.. Find the volume of the solid domain enclosed by the two paraboloids z = (x + y ) and z = 1 + x + y Compute the volume of the ellipsoid E : x a + y b + z c Find the volume of the ice-cream cone x + y + z 1, z x + y, by using spherical coordinates. 4. Line integrals 4.1. Let F = yi+xj x +y and c(t) = (R cos t, R sin t) a parametrization of the circle of radius R in R. a) Compute F. b) Compute c F ds. c) How do you reconcile the results obtained at a) and b)? 4.. Exercise 18 (p. 449) 4.3. Determine the work done by the gravitational vector field when a point of mass m moves from P (5, 4, 1) to Q(0, 1, 0) Compute sin(πx)dy cos(πy)dz where c is the oriented triangle P QRP with c P (1, 0, 0), Q(0, 1, 0) and R(0, 0, 1) Compute c xyz dx + x z dy + x y dz where c is *any* path joining P (1, 1, 1) to Q(1,, 4) A particle follows the path c(t) = ( cos(t), sin(t), 3t). Find the length of the path travelled by the particle between t = 0 and t = π/ Let F (x, y, z) = (x + y )i + (xy + 3y )j. a) Prove that curl F = 0. b) Argue the existence of a scalar function such that f = F. c) Find the function f. d) Evaluate the integral C F, where C is the arc of the curve y = sin 3 x from (0, 0) to (π/, 1) in the xy-plane.

3 PRACTICE PROBLEMS 3 5. Surface integrals 5.1. Compute the surface area of a surface of revolution. 5.. Compute the area of the top of the ice-cream cone: x + y + z = 1, z x + y The cylinder x + y = x cuts out a portion of a surface Σ from the upper nappe of the cone x + y = z. Compute Σ (x4 y 4 + y z z x + 1) ds Find the average of x over the sphere of radius R (centered at the origin) Same question as above, but replace the sphere with the ellipsoid x a + y b + z c = Parametrized surfaces 6.1. Find the equation of the tangent plane to surface x = u v, y = u + v, z = u + 4v at ( 1/4, 1/, ). 6.. Find the unit vector normal to the surface x = 3 cos θ sin φ, y = sin θ sin φ, z = cos φ for θ [0, π] and φ [0, π] Parametrize the hyperboloid x + y z = 5 with x = 5 cosh u cos θ, y = 5 cosh u sin θ, z = 5 sinh u. Find equation of the tangent plane at (3, 4, 0) Let Σ be the surface parametrized by Φ(u, v) = (u, u 3 cos v, u 3 sin v) with (u, v) [0, 1] [0, π]. Calculate the normal vector N(u, v) and the area of the resulting surface Parametrize the surface obtained through the intersection of the plane x + y + z = 1 and the cylinder x + y = 1. Use this parametrization to compute the area of the surface Rotating a parametrized curve γ(t) = (c 1 (t), c (t)), where a t b, around the x-axis defines a surface of revolution parametrized by Φ(u, v) = (c 1 (u), c (u) cos v, c (u) sin v), a u b, 0 θ π Calculate the normalized unit vector defined by this parametrization, when the original curve is the cycloid defined by γ(t) = (t sin t, 1 cos t) 7. Flux 7.1. Let Σ the hemisphere x + y + z = 1, z 0 and F = xi + yj. a) Compute the flow of F through Σ using spherical coordinates. b) Compute the flow of F through Σ by realizing Σ as the graph of the function z = 1 x y.

4 4 PRACTICE PROBLEMS 7.. Let T the triangle with vertices A = (1, 0, 0), B = (0, 1, 0) and C = (0, 0, 1). a) Compute the flow of the position vector r through T by realizing T as the graph of z = 1 x y above a triangle D in the xy-plane. b) Do the same computation by parametrizing T as follows: P = (1 u)a + u ((1 v)b + vc) = (1 u, u(1 v), uv) =: Φ(u, v) where P is point in T and 0 u 1, 0 v A fluid (of density one unit) has velocity vector field F = xi (x + y)j + zk. Compute the mass of the fluid flowing through the hemisphere Σ = {(x, y, z) : x + y + z = 1, z 0} per unit time. b) Do the same computation with Σ including the planar base of the hemisphere The plane x + y + z = 1 cuts the unit sphere x + y + z = 1 into two parts. Let Σ the upper part, i.e. the region of the unit sphere where x + y + z 1. (Exercise 3 of 8.3). a) Parametrize the surface Σ. b) Parametrize the boundary Σ compatible with the outward unit vector n = r on Σ. c) Let F = r (i + j + k) Compute the flow of F through Σ by using Stokes formula and the integrate on the boundary Σ. e) Compute the flow of F through Σ by using Stokes formula to notice that the flow is the same as the flow through D, where D is the disk cut out by the plane x + y + z = 1 through the unit ball x + y + z 1. Hint for parametrization: first notice that Σ is a circle of radius 3 centered at P ( 1 3, 1 3, 1 3 ). The unit normal vector perpendicular on the plane x + y + z = 1 is the vector k = i+j+k 3. A = (0, 1, 0) is one of the points on Σ. Let j the normalized vector in the direction of P A, that is j = i+j k 6. Let now i = j k = i k. Then we can parametrize Σ by using spherical coordinates in the system {i, j, k }. If one wants to parametrize Σ without going through Σ first, let θ the angle made by i and P M. Then P M is a vector of length 3 in the plane spanned by i and j hence P M = 3 (cos θi + sin θj ). Since P M = M P we arrive at M = ( 1 3, 1 3, 1 ( cos θ 3 ) + sin θ 3 3, sin θ, cos θ sin θ ) and this is a parametrization of Σ a) Compute r ds where H is the spiral ramp defined by the parametrization H Φ(u, v) = (u cos v, u sin v, v), 0 u 1, 0 v π b) Compute the line integral H r Compute C F, where C is the curve defined by the intersection of the plane y + z = and the cylinder x + y = 1, and F = y i + xj + z k.

5 PRACTICE PROBLEMS 5 8. Curl and Stokes formula 8.1. Stokes formula. S = surface, n S unit normal vector to S, S the boundary of S (orientatation compatible with the choice of the unit normal vector n S, and F an arbitrary vector field. Then: ( F ) ds = F S 8.. a) Use Stokes formula to prove that the work of the gravitational vector field G = GMm r r 3 around any closed curve C in R3 (not passing through the origin) is zero. b) Can you prove a) without using Stokes theorem? 8.3. Use Stokes formula to argue that any irrotational vector field in R 3 (0, 0, 0) (that is, a vector field in R 3 which has at most a singularity at the origin) is the gradient of the vector field. Note that this property does not hold for R (0, 0) instead of R 3 (0, 0, 0) (see exercise 1/8.3 of homework). Try to explain this difference Memorize the formulas for dv and ds in spherical coordinates: on the ball of radius R centered at the origin, dv = ρ sin φ dρdθdφ (the Jacobian in spherical coordinates) on the sphere of radius R, ds = R sin φdθdφ Let v a fixed vector in R 3, and F = v a constant vector field. i) Prove that there exists a vector field G such that F = G. ii) Find such a vector field G Let S a closed (oriented) surface in R 3 and a a fixed vector in R 3. Prove that a ds = 0. (Hint: can you use the previous exercise?) S 8.7. Let H the spiral ramp x = u cos θ, y = u sin θ, z = θ, 0 u 1, 0 θ π a) Compute r ds, where r is the position vector field. (Assume the normal H vector n H is in the direction of T u T θ.) b) Compute the line integral H r Let S the sphere of radius R centered at (1,, 3). Find the unit normal vector n S of S Find the integral of F = x i + y j zk around the triangle with vertices (0, 0, 0), (0,, 0) and (0, 0, 3) (in this order) using Stokes formula Evaluate C F where C is the curve defined by the intersection of the plane y + z = and the cylinder x + y = 1, and F = ( y, x, z ) Verify Stokes theorem for the vector field F = z i xj+y 3 k over the upper hemisphere of the unit sphere x + y + z = 1. S

6 6 PRACTICE PROBLEMS 8.1. Let G = y zi + (x y + z 3z)j + (yz + e z )k. a) Use Stokes theorem to express the line integral C G as a surface integral, where C denotes the piecewise linear (square) contour that goes from the origin to (0,, 0), then to (0,, ), then to (0, 0, ) and back to the origin. b) Evaluate the integral. (Hint: do not evaluate the line integral directly, unless you have time and want to double check your answer.) Use Stokes theorem to compute Σ ( F ) ds, where F = yzi+yzj+xyk and Σ is the part of the sphere x + y + z = 4 which lies inside the cylinder x + y = 1 and above the xy-plane Verify Stokes theorem for the vector fields F = z i xj + y 3 k over the upper hemisphere of the unit sphere x + y + z = Exercises 4,5, 7, 16-(p.606). 9. Green s formula 9.1. Green s formula. Let D a domain in R and the oriented (counterclockwise) curve C its boundary. Let P (x, y) and Q(x, y) two functions of two variables. Use Stokes formula to prove that ( Q P dx + Qdy = x P ) dxdy y C 9.. Use Green s theorem to evaluate the contour integral (1 + y 8 )dx + (x + e y )dy C where C is the boundary of the domain enclosed by the curve y = x and the lines x = 1 and y = Use Green s formula to compute the area of the interior of the ellipse x a + y b = 1. (Hint: choose P (x, y) = y and Q(x, y) = x Use Green s theorem to find the area bounded by one arc of the cycloid x = θ sin θ, y = 1 cos θ, 0 θ π and the x-axis. (The cycloid is drawn on p. 144 of the textbook.) D 10. Divergence and Gauss Formula Gauss Theorem. Let W a solid domain in R 3 and W its oriented boundary surface. Let F an arbitrary vector field (defined on W ). Then div F dxdydz = F n da W where n is the unit normal vector (pointing outward) on W and ds the surface element on W. W

7 PRACTICE PROBLEMS a) State Gauss theorem. Explain briefly what each symbol in the theorem stands for. (Assume all the differentiability that you want.) b) Use Gauss theorem to compute S F ds, where F = xyi + (y + e xz )j + sin(xy)k and S is the boundary surface of the region W bounded by the parabolic cylinder z = 1 x and the planes z = 0, y = 0, and y = 5. r Let F = r 3 and assume S a closed surface in R3 enclosing a solid region W. Prove that S F ds = {4π, if (0, 0, 0) W 0, otherwise Assume that the points p 1,..., p k in R 3 have electric charges q 1,..., q k. The electric potential (by Coulomb s law) is given by the scalar function φ(x) = 1 4π k i=1 q i x p i while the electric field is given by E = φ. Use Gauss law to compute the flux of E through a closed surface Σ Let p 1, p,..., p 8 eight different points in R 3. Consider the vector field F (x) = 1 4π 10 k=1 10 k x p k x p k 3 If Σ is a closed surface (not passing through any of the given eight points) such that Σ F ds = 11010, which of the eight points lie inside of Σ? Let B(r) the solid ball of radius r centered at x 0 R 3 and F an arbitrary 1 vector field. Prove that lim r 0 vol(b(r)) B(r) F ds = div F (x0 ) Let Ω the box [ 1, 1] [ 1, 1] [ 1, 1]. Compute the total flux of the vector field F = xyi + (y y )j + (x y + z)k out the cube Ω Read pages in the textbook on curl and divergence, and on Gibbs interpretation of divergence Let F = x i+(x y xy)j x zk. Does there exist G such that F = G? Ex. 14, 17, 18 (p. 575)

8 8 PRACTICE PROBLEMS 11. Vector fields Definitions: F is conservative if there exist φ(x, y, z) a scalar function (potential) such that F = φ. Note that the work of a conservative field along a curve in R 3 depends solely on the endpoints of the curve (fundamental theorem of calculus). Equivalently, the work (or the circulation) of a conservative vector field along a closed curve is always zero. F is irrotational if F = 0. F is incompressible if div F = 0 (also called divergence-free) Give an example of an irrotational vector field Give an example of a divergence-free vector field Let φ(x, y, z) a real-valued function. Prove that the vector field φ is irrotational Prove that the gravitational vector field G = GMm r is irrotational. r Let a a fixed vector and F = a r. Is F conservative (irrotational)? If so, find a potential for it Prove that (r n ) = nr n r Show that a vector field which is both irrotational and incompressible is of the form F = φ where the scalar function φ(x, y, z) satisfies Laplace equation φ x + φ y + φ z = Show that the vector field in R : F = xi yj is irrotational and incompressible. Determine a potential φ for F. What are the level curves of φ? Prove that there exists no vector field G such that curl G = xi + 3yzj xz k Let E = E 1 i + E j + E 3 k, F = F 1 i + F j + F 3 k G = G 1 i + G j + G 3 k three vector fields. Recall the determinant formula E 1 E E 3 E ( F G) = F 1 F F 3 G 1 G G 3