Chapter 16 review

Vectors

100

What constitutes a vector field F as conservative ?

If there exists a function F = grad f

(If it also has a gradient vector field for some function)

200

Sketch the Vector field; 

F(x,y) = <x², 2y>

(For images, refer to slides) 

Vectors pointing inwards to the x-axis and origin when x < 0, and outwards when x > 0. Also, vector magnitudes are increasing while going further from the origin.

300

Which one of these vector fields are conservative. (select all that apply)

a. F(x,y,z) = (xy + y^2)i + (x^2 + 2xy)j

b. F(x,y,z) = (ye^x + sin y)i + (e^x + xcos y)j

c. F(x,y,z) = (ln y + y/x)i + (ln x + x/y)j

b, c

400

Find the curl and divergence of the vector field:

F(x,y,z) = ⟨cos(xπ),sin(yπ),-xyz⟩

curlF= ⟨−xz,yz,0⟩ 

divF = -πsin(xπ) + πcos(yπ) - xy 

500

Fill in these blanks for vector definitions: (Suppose F is continuous vector field along some domain D)

A path C is …                                   

___ if it doesn’t cross itself

___if it’s initial and final points are same

A region D is…

___if it doesn’t contain any of its boundary points

___if we can connect any two points in the region with a path lying only in D

___if it is connected and has no holes

A path C is …                                   

simple if it doesn’t cross itself

closed if it’s initial and final points are same

A region D is…

Open if it doesn’t contain any of its boundary points

Connected if we can connect any two points in the region with a path lying only in D

Simply-connected if it is connected and has no holes

Line Integrals

100

Given: 

∫f(x,y)dS

Where x and y are functions of t

What does dS represent?

root ((dx/dt)² + (dy/dy)²) dt

200

Does Fundamental Theorem of Calculus apply to this vector field? If yes, find the potential function.

⟨cos(x)e^sin(x) - 3z, z/(√(1-y²)), sin^-1(y) - 3x⟩

Yes 

e^sin(x) + (z*sin^-1(y)) - 3xz)

300

Compute the work type integral

∫f ·dr, F(x,y) = ⟨xy, 3y²⟩, r(t) = ⟨11t⁴, t³⟩

 0<t<1

45

Using F(r(t)) · r’(t)

400

Evaluate the work type integral from (0,0,0) to (1,1,1)

∫f ·dr

f(x,y,z) = ⟨2x + yz, xz, xy⟩

r(t) = ⟨t, t³, t²⟩

2

Using fundamental theorem

Potential function: x² + xyz

Evaluate from t=0 to t=1 

500

Evaluate the line integral from segments from (0,0) to (4,0) and (4,0) to (5,2) 

∫(x^2 + y)dx + (xy)dy

Parametrize twice 

1.) 0<t<4 x = t, y = 0 dx=dt, dy=0

2.) 4<t<5 x = t, y = 2t-8 dx=dt, dy=2dt

Integral 1: from 0 to 4: t^2 dt

Integral 2: from 4 to 5: t^2 + 2t-8 +2t^2 -8t

Add them: 142/3

Green's and Divergence Theorem

100

Green’s and the Divergence Theorem are generalizations of what?

The Fundamental Theorem of Calculus

200

What condition does Green’s Theorem need?

That the curve is closed.

300

The Divergence Theorem states that the flux surface integral is equivalent to what?

Flux volume integral

400

Set up the Divergence Theorem (solve for integrand and bounds) of: F(x)=〈sin(πx), zy3, z2+4x〉, x:[-1, 2], y:[0, 1], z:[1, 4]

∫₁⁴∫₀¹∫_-1²πcos(πx)+3zy²+2z dxdydz

500

Solve ∫_Cyx²dx - x²dy where C: the bottom half x²+y²=25 and y=0, going counter clockwise

-(625/8)π

Surface Integral

100

Which type of surface integral is this example? Mass or Work type?

∫∫sin(x²+y²)√(1+4x²+4y²)dA

Mass type

200

You cannot integrate flux across this type of surface (Example: Mobius strip)

Non Orientable surface

300

Given the field F and the surface integral,

∫∫ F ·〈1/√3, 1/√3, 1/√3〉dS

What can you infer about the surface S?

The surface is a plane, 

<not required> with equal direction cosines on its normal vector

400

Complete the Integral

(but don't evaluate)

r(u,v) =〈u² + v, v² + u, 3〉

f(x(u,v),y(u,v),z(u,v)) = (u + 1)/(e^v)

∫∫f(x,y,z)dS = 

∫∫(u + 1)/(e^v) (Missing part)  dudv

(4uv - 1)

∫∫(u + 1)/(e^v) * (4uv - 1)dS

500

Set up (but don't evaluate) a flux integral where 

F =〈xy, xz, yz〉and the surface is 

x = sin(y)  

(oriented in the positive x direction)

∫∫ysin(y) - zsin(y)cos(y)dydz

Stokes' Theorem

100

When stokes theorem integrand is 0, then the function is said to be:

Conservative

200

F(x,y,z) = xi + yj + zk is stokes theorem possible?

Yes

300

What is the most convenient to use as the surface curve if using stokes to compute:

∫∫curlF·dS where S is the part of the sphere x² + y² + z² = 4 that lies inside the cylinder x² + y² = 1

The circle given by x² + y² =1 (where z = √3)

400

DAILY DOUBLE

Use Stokes’ Theorem to evaluate ∫∫curlF·dS 

F =〈x², 2x, cos(√(x²+ y²))〉

Surface: z = sin(√(x²+ y²))

x²+ y² <= π²

^{assume CCW orientation}

2π³

500

Use Stokes’ Theorem to evaluate ∫∫curlF·dS where 

F =〈z²-1, z+xy³, 6〉and S: x=6-4y²-4z² in front of x=-10

assume CCW orientation

4π