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Final Exam Review Pack
1 Lecture 23: Vector Fields
Summary of Lecture
- A vector field is a vector function of position. That is, a vector field is a function from R^2 (2 dimensional) or R^3 (3 dimensional) that assigns each point a vector 〈f (x, y), g(x, y)〉 (in 3 dimensions it takes the form 〈f (x, y, z), g(x, y, z), h(x, y, z)〉). We call the functions f, g, h the component functions of the vector field.
- The domain of the vector field is the intersection of the domains of component functions. A vector field is continuous/differentiable if and only if all the components are continuous/differentiable. Vector fields are used in many applications in physics (gravitational, electricity, velocity fields) and other fields. In 2 and 3 dimensional setting, vector fields can be visualized as a set of arrows in 2 or 3 space.
- A radial vector field has the form F = C (^) ‖rr‖p where p and r are constants. Here r = 〈x, y〉 in 2 dimensions and r = 〈x, y, z〉 in 3 dimensions.
- A potential function for a vector field F is a scalar function φ such that F = ∇φ.
- A vector field F is said to be a conservative vector field if F has a potential function φ
- A vector field F = 〈F 1 , F 2 , F 3 〉 satisfies the cross partial condition (equivalently, irrotational ) if ∂F 2 ∂x =^
∂F 1
∂y
∂F 3
∂y =^
∂F 2
∂z
∂F 1
∂z =^
∂F 3
∂x
- All conservative vector fields satisfy the cross partial condition. But the converse is not true. That is, there are non conservative vector fields that satisfy the cross partial condition.
- If the domain of the vector field F is simply connected, then F is conservative if and only if it satisfies the cross partial condition.
Exercises
- Is the vector field F = 〈yz, xz, y〉 conservative?
- Find a potential function to the vector field F = 〈x, y〉
- Find a potential function to the vector field F = 〈yexy,xexy^ 〉
- Find a potential function to the vector field F = 〈yz^2 , xz^2 , 2 xyz〉
- Find a potential function to the vector field F = 〈 2 xzex^2 , 0 , ex^2 〉
- Find a potential function φ(x, y) for F = (^) ‖rr‖ 3 and a potential function ψ(x, y, z) G = (^) ‖rr‖ 4 in R^3.
- Which of (A) or (B) in the following figure is the contour plot of a potential function for the vector field F. Recall that the gradient vectors are orthogonal to the level curves.
Answers
- Not conservative.
- φ(x, y) =^12 x^2 +^12 y^2
- φ(x, y) = exy
- φ(x, y, z) = xyz^2
- φ(x, y, z) = zex^2
- φ = − (^2) ‖^1 r‖ 2 and ψ = − (^3) ‖^1 r‖ 3
- (B)
Exercises
- Let f (x, y, z) = x + yz and let C be the line segment from P = (0, 0 , 0) to (6, 2 , 2). (a) Calculate f (c(t)) and ds = ‖c′(t)‖dt for the parametrization c(t) = 〈 6 t, 2 t, 2 t〉 for 0 ≤ t ≤ 1. (b) Evaluate
C f (x, y, z) ds.
- Let F = 〈y^2 , x^2 〉 and let C be the curve y = x−^1 for 1 ≤ x ≤ 2 oriented from left to right. (a) Calculate F(c(t)) and ds = c′(t) dt for the parametrization of C given by c(t) = 〈t, t−^1 〉. (b) Calculate the dot product F(c(t)) · c′(t) and evaluate
C^ F^ ·^ dS.
- Compute the integral of the scalar function f (x, y, z) = x^2 + y^2 + z^2 on the curve r(t) = 〈cos(t), sin(t), t〉 for t ∈ [0, π]
- Compute the line integral of the vector field F = 〈x, y, z〉 on the curve r(t) = 〈cos(t), sin(t), t〉 for t ∈ [0, π].
- Compute the integral of the scalar function f (x, y) = √1 + 9xy on the curve y = x^2 for 0 ≤ x ≤ 1.
- Compute the line integral
C^ f ds^ of the function^ f^ (x, y) =^
y^3 x^7 on the curve^ y^ =
4 s
(^4) for 1 ≤ x ≤ 2.
- Compute the line integral
C f ds of the function f (x, y, z) = xez^2 on the piecewise linear path from (0, 0 , 1) to (0, 2 , 0) to (1, 1 , 1).
- Compute the line integral
C^ f ds^ of the function^ f^ (x, y, z) =^ x
(^2) z on the curve given by r(t) = 〈et, √ 2 t, e−t〉, 0 ≤ t ≤ 1
- Calculate
C^1 ds, where the curve^ C^ is parametrized by^ r(t) =^ 〈e
t, √ 2 t, e−t〉 for t ∈ [0, 2].
- Compute the vector line integral
C^ F^ ·^ ds^ of the vector field^ F^ =^ 〈x
(^2) , xy〉 on the line segment from (0, 0) to (2, 2).
- Compute the vector line integral
C^ F^ ·^ ds^ of the vector field^ F^ =^ 〈^4 , y〉^ on the quarter circle^ x
(^2) + y (^2) = 1 with x ≤ 0 and y ≤ 0 oriented counterclockwise.
- Compute the vector line integral
C^ F^ ·^ ds^ of the vector field^ F^ =^ 〈x
(^2) , xy〉 on the part of the circle x (^2) + y (^2) = 9 with x ≤ 0, y ≥ 0 oriented clockwise.
- Evaluate the integral
C y dx + z dy + x dz on the curve parametrized by c(t) = 〈2 + t−^1 , t^3 , t^2 ) for o ≤ t ≤ 1.
- Calculate the line integral of F = 〈ez^ , ex−y^ , ey^ 〉over the blue path from P to Q on the figure below.
- Calculate the work done by the force field F = 〈x, y, z〉 along the path r(t) = 〈cos(t), sin(t), t〉 from 0 ≤ t ≤ 3 π.
Answers
- (a) ds = 2√ 11 dt (b) 26
(a) ds = 〈 1 , −t−^2 〉dt (b) − (^12)
√2(π + π^3 /3)
π 4 4
(^145)
576 1 (65^3 /^2 − 23 /^2 )
2 (e^ −^ 1)
- 12 (e^2 + 1)
- e^2 − e−^2
- (^163)
- 0
- (^4110)
- 2 − e−^1 − e
- 9 π
2 2
Exercises
- Evaluate the line integral of F = 〈 3 , 6 y〉 over the path given by r(t) = 〈t, 2 t−^1 〉 for 1 ≤ t ≤ 4.
- Evaluate the line integral of F = 〈yez^ , xez^ , xyez^ 〉 over the curve c(t) = 〈t^2 , t^3 , t − 1 〉 for 1 ≤ t ≤ 2.
- Find a potential function for F = 〈z, 1 , x〉.
- Find a potential function for F = 〈x, y〉 or determine it’s not conservative.
- Find a potential function for F = y^2 i + (2xy + ez^ )j + yez^ k or determine it is not conservative
- Find a potential function for F = 〈cos(xz), sin(yz), xy sin(z)〉 or determine it is not conservative.
- Find a potential function for F = 〈yzexy^ , xzexy^ − z, exy^ − y〉.
- Evaluate ∫ C 2 xyz dx + x^2 z dy + x^2 y dz over the path c(t) = 〈t^2 , sin(πt/4), et^2 −^2 t〉 for 0 ≤ t ≤ 2.
- Evaluate ∮ C sin(x) dx + z cos(y) dy + sin(y) dz Where C is the ellipse 4x^2 + 9y^2 = 36, oriented clockwise.
- Let F =
x ,^ −^
y
. Calculate the work against F required to move an object from (1, 1) to (3, 4) along any path in the first quadrant.
- The vector field F =
〈 (^) x x^2 + y^2 ,^
y x^2 + y^2
is defined on the domain D = {(x, y)|x, y 6 = 0}. (a) Is D simply connected? (b) Show that F satisfies the cross partial condition. Does this guarantee that F is conservative? (c) Show that F is conservative by finding a potential function. (d) Does these results contradict the highlighted statement 7 in Lecture 23 summary?
- The vector field F =
〈 (^) −y x^2 + y^2 ,^
x x^2 + y^2
is defined on the domain D = {(x, y)|x, y 6 = 0}. (a) Show that F satisfies the cross partial condition. Does this guarantee that F is conservative? (b) Is F conservative? (c) Does these results contradict the highlighted statement 7 in Lecture 23 summary?
Answers
- − (^94)
- 32e − 1
- V (x, y, z) = xz + y
- Not Conservative
- V (x, y, z) = y^2 z + ez^ y
- Not Conservative
- V (x, y, z) = zexy^ − yz
- 16
- 0
- ln 4 − ln 3
- (a) no (b) can easily verify (c) V (x, y) =^12 ln(x^2 + y^2 ) (d) No, conservative vector fields can be defined in non simply connected regions.
- (a) Can easily verify (b) No (c) No, because the domain is not simply connected.
Exercises
- Let G(x, y) = 〈x, y, xy〉 (a) Calculate Gx, Gy and n(x, y) (b) Let S be the part of the surface with parameter domain D = {(x, y) : x^2 + y^2 ≤ 1 , x ≥ 0 , y ≥ 0 }, verify the following formula and evaluate using polar coordinates. x S
1 dS =
x D
√1 + x (^2) + y (^2) dx dy
(c) Verify the following formula and evaluate x S
z ds =
∫ (^) π/ 2 0
0 (sin(θ) cos(θ))r^3 √1 + r^2 dr dθ
- Find the normal vector to the surface G(u, v) = 〈 2 u + v, u − 4 v, 3 u〉 at the point u = 1, v = 4. Also find the equation of the tangent plane to G at that point.
- Find the normal vector to the surface G(r, θ) = 〈r cos(θ), r sin(θ), 1 − r^2 〉 at the point r = 1/2, θ = π/4. Also find the equation of the tangent plane to G at that point.
- Calculate the scalar surface integral s S f (x, y, z) dS for the function f (x, y, z) = x(x^2 + y^2 ) on the surface G(u, v) = 〈u cos(v), u sin(v), u〉 where 0 ≤ u ≤ 1, 0 ≤ v ≤ 1
- Calculate the scalar surface integral s S f (x, y, z) dS for the function f (x, y, z) = √x^2 + y^2 on the surface G(r, θ) = 〈r cos(θ), r sin(θ), θ〉 where 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2 π
- Calculate the scalar surface integral s S f (x, y, z) dS for the function f (x, y, z) = 1 on the surface y = 9 − z^2 where 0 ≤ x ≤ z ≤ 3
- Calculate the scalar surface integral s S f (x, y, z) dS for the function f (x, y, z) = z on the part of the plane x + y + z = 1, where x, y, z ≥ 0
- Calculate the scalar surface integral s S f (x, y, z) dS for the function f (x, y, z) = z^2 on part of the plane x + y + z = 0 contained in the cylinder x^2 + y^2 = 1
- Calculate the scalar surface integral s S f (x, y, z) dS for the function f (x, y, z) = z on the part of the surface z = x^3 where 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
- Calculate the scalar surface integral s S f (x, y, z) dS for the function f (x, y, z) = xy +ez^ on the triangle shown below.
- Find the surface area of the portion S if the cone z^2 = x^2 + y^2 where z ≥ 0, contained within the cylinder y^2 + z^2 ≤ 1.
Answers
- (a) Gx = 〈 1 , 0 , y〉, Gy = 〈 0 , 1 , x〉 n(x, y) = 〈−y, −x, 1 〉 (b) (
√ 2 − 1)π 6 (c)
〈 12 , 3 , − 9 〉 Plane 4x + y − 3 z = 0
〈 2 r^2 cos θ, 2 r^2 sin θ, r〉 Plane 2√ 2 x + 2√ 2 y + 4z = 5
2 π(
- π
- 3e^3 − 6 e^2 + 3e + 1
- π
Exercises
- Compute the flux
x S
F · dS for the function F = 〈y, z, x〉 on the plane 3x − 4 y + z = 1 with upward pointing normal , where x, y ∈ [0, 1]
- Compute the flux x S
F · dS for the function F = 〈ez^ , z, x〉 on the surface parametrized by G(r, s) = 〈rs, r+s, r〉 oriented by Gr × Gs, where r, s ∈ [0, 1]
- Compute the flux x S
F · dS for the function F = 〈 0 , 3 , x〉 on the part of the sphere of radius 3 on the first octant with outward pointing normal.
- Compute the flux
x S
F · dS for the function F = 〈x, y, z〉 on the part of the unit sphere with outward pointing normal, where 1/ 2 ≤ z ≤ √ 3 / 2
- Compute the flux
x S
F · dS for the function F = 〈z, z, x〉 on the surface z = 9 − x^2 − y^2 with upward pointing normal, where x, y, z ≥ 0
- Compute the flux
x S
F · dS for the function F = 〈sin(y), sin(z), yz〉 on the rectangle 0 ≤ y ≤ 2, 0 ≤ z ≤ 3 in the yz plane, normal pointing in the negative x direction.
- Compute the flux x S
F · dS for the function F = 〈y^2 , 2 , −x〉 on the portion of the plane x + y + z = 1 with upward pointing normal in the first octant.
- Compute the flux x S
F · dS for the function F = 〈x, y, ez^ 〉 on the cylinder x^2 + y^2 = 4 with outward pointing normal , where 1 ≤ z ≤ 5
- Compute the flux
x S
F · dS for the function F = 〈xz, yz, z−^1 〉 on the disk of radius 3 at height 4 parallel to the xy plane, with upward pointing normal.
- Compute the flux
x S
F · dS for the function F = 〈xy, y, 0 〉 on the cone z^2 = x^2 + y^2 , where x^2 + y^2 ≤ 4, z ≥ 0 with downward pointing normal
- Compute the flux
x S
F · dS for the function F = 〈 0 , 0 , ey+z^ 〉 on the boundary of the unit cube 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 , 0 ≤ z ≤ 1.
- Compute the flux
x S
F · dS for the function F = 〈 0 , 0 , z^2 〉 on the helicoid parametrized by G(u, v) = 〈u cos(v), u sin(v), v〉 , where 0 ≤ u ≤ 1 and 0 ≤ v ≤ 2 π with upward pointing normal.
Answers
- − 4
- 4/ 3 − e
- 2712 (3π + 4)
- π(1 − √3)
- 3(cos 2 − 1)
- (^1112)
- 32π
- 94 π
- 83 π
- (e − 1)^2
- 4 π 3 3
- Let x^3 + y^3 = 3xy be the follum of Descartes
(a) Show that the follum has a parametrization in terms of t = y/x given by x = (^) 1 +^3 t t 3 , y = 3 t 2 t ∈ (−∞, ∞), t 6 == − 1 1 +^ t^3 (b) Show that x dy − y dx = 9 t
2 (1 + t^3 )^2 dt (c) Find the area of the loop of the follum Answers
- 0
- e^2 / 2 − e^3 / 3 − 1 / 6
- −π/ 4
- − 1 / 30
- 1/ 6
- − 30
- (e (^2) − 1)(5 − e (^4) ) 2
- 1
- 9π
- 1/ 2
- 3π
- 8/ 3
- (c) : 3/ 2
7 Lecture 29: Stoke’s Theorem
Summary of Lecture
- The Curl of a vector field F = 〈F 1 , F 2 , F 3 〉 is given by the symbolic cross product:
CurlF = ∇ × F =
i j k ∂ ∂x
∂y
∂z F 1 F 2 F 3
The symbol ∇ =
∂x ,
∂y ,
∂z
is called the del operator. A vector field F is called irrotational if ∇ × F = 0
- Conservative vector fields are irrotational, but not all irrotational vector fields are conservative. But an irrotational vector field defined on a simply connected domain is always conservative.
- The Stoke’s Theorem is used mainly in 2 cases (a) To evaluate the flux of the curl of a certain vector field by converting it to a vector line integral. (b) To evaluate a vector line integral (usually a union of lines having different parametrizations, like a tri- angle) as a single vector surface integral. Also, if we can find a vector potential G to a vector field F (so that F = ∇ × G), then we can evaluate the flux of F across any surface as a line integral using the Stoke’s theorem. (c) Stoke’s Thoerem Let S be a smooth oriented surface in R^3 , with a (piecewise) smooth closed boundary curve ∂S whose orientation is consistent with that of S. Assume that F = 〈F 1 , F 2 , F 3 〉 is a vector fields whose componets are continuously differentiable on S, then ∮ ∂S^ Fdr^ =
x S
(∇ × F) · n ds
In particular, if S is a closed surface, then s S (∇ × F) · n ds = 0
Exercises
- Find ∇ × F for F = 〈z − y^2 , x + z^3 , y + x^2 〉
- Find ∇ × F for F =
〈 (^) y x ,
y z ,
z x
- Find ∇ × F for F = 〈ey^ , sin(x), cos(x)〉
- Find ∇ × F for F =
〈 (^) x x^2 + y^2 ,^
y x^2 + y^2 ,^0
- Compute the flux of curl(F) of the vector field F = 〈ez^2 − y, ez^2 + x, cos(xz)〉 on the upper hemesphere using Stoke’s Theorem.
Answers
- 〈 1 − 3 z^2 , 1 − 2 x, 1 + 2y〉
- 〈y/z^2 , z/x^2 , − 1 /x〉
- 〈 0 , sin(x), cos(x) − ey^ 〉
- 〈 0 , 0 , 0 〉
- 2π
- 1/ 2
- (a) A = 〈 0 , 0 , ey^ − ex〉 (b) 0 (c)π/ 2
- 234
- 32
- 3π/ 4
8 Lecture 30: The Divergence Theorem
Summary of Lecture
- The divergence of a vector field F = 〈F 1 , F 2 , F 3 〉 is given by the symbolic dot product div(F) = ∇ · F = ∂F ∂x^1 + ∂F ∂y^2 + ∂F ∂z^3 This results in a scalar valued function. If for a vector field F, div(F) = 0, we say F is incompressible.
- For any vector field F with continuous second order partial derivatives, ∇ · (∇ × F) = 0. However, if for a given vector field G such that div(G) = 0, G = curl(F) for some vector field F only if G is defined on a simply connected region.
- The divergence Theorem is used to evaluate a vector surface integral (flux) over a closed, bounded surface S, by converting it to a triple integral.
- The divergence Theorem: Let F be a vector field defined on a connected, simply connected region D in R^3 enclosed by a smooth oriented surface S. Also assume that the components of F are continuously differentiable. Then (^) x S
F · n ds = y D
(∇ · F)dV Where n is the outward unit normal vector to S. Exercises
Use the divergence Theorem to evaluate the following surface integrals for the function F given on the surface S.
- F = 〈 0 , 0 , z^3 / 3 〉 where S is the sphere x^2 + y^2 + z^2 = 1
- F = 〈y, z, x〉 where S is the sphere x^2 + y^2 + z^2 = 1
- F = 〈x^3 , 0 , z^3 〉 S is the first octant of the sphere x^2 + y^2 + z^2 = 4
- F = 〈ex+y^ , ex+z^ , ex+y^ 〉 where S is the boundary of the unit cube x, y, z ∈ [0, 1]
- F = 〈x, y^2 , z + y〉 where S is the boundary of the region contained in the cylinder x^2 + y^2 = 4 between the planes z = x and z = 8
- F = 〈x^2 − z^2 , ez^2 − cos(x), y^3 〉 where S is the boundary of the region bounded by x + 2y + 4z = 12 and the coordinate planes in the first octant.
- F = 〈x + y, z, z − x〉 where S is the boundary of the region between the paraboloid z = 9 − x^2 − y^2 and the xy-plane.
- F = 〈ez^2 , 2 y + sin(x^2 z), 4 z + √x^2 + 9y^2 〉 where S is the region x^2 + y^2 ≤ z ≤ 8 − x^2 − y^2
- Calculate the flux of the vector field F = 2xyi − y^2 j + k through the surface S in the following figure.
