Here are the assigned readings and exercises (Reading 1, Reading 2, Reading 3, Reading 4, Reading 5, Reading 6, Reading 7, Reading 8, Reading 9, Reading 10, Reading 11, Reading 12, Reading 13, Reading 14, Reading 15, Reading 16, Reading 17, Reading 18, Reading 19, Reading 20, Reading 21, Reading 22, Reading 23, Reading 24, Reading 25, Reading 26, Reading 27, Reading 28, Reading 29, Reading 30); but anything whose assigned date is in the future is subject to change!

- Vectors:
- Date assigned: March 28 Wednesday.
- Date due: March 29 Thursday.
- Reading from the textbook: As needed: Review §§11.1–11.5.
- Reading from my notes: Optional: Page 2 through the top of page 16 (§§1.1–1.13).
- Online notes: Required: Vector operations.
- Exercises due:
- Give a formula
for the vector
from the point (
*x*_{1},*y*_{1}) to the point (*x*_{2},*y*_{2}). - If
**u**and**v**are vectors in 2 dimensions, then is**u**×**v**a scalar or a vector? - If
**u**and**v**are vectors in 3 dimensions, then is**u**×**v**a scalar or a vector?

- Give a formula
for the vector
from the point (

- Parametrized curves:
- Date assigned: March 29 Thursday.
- Date due: April 2 Monday.
- Reading from the textbook: Pages 642–648 (§12.1).
- Reading from my notes: Page 16 and the first paragraph of page 17 (the first half of §1.14).
- Exercises due:
- If
*C*is a point-valued function, so that*P*=*C*(*t*) is a point (for each scalar value of*t*), then what type of value does its derivative*C*′ take?; that is, is d*P*/d*t*=*C*′(*t*) a point, a scalar, a vector, or what? - If
**c**is a vector-valued function, so that**r**=**c**(*t*) is a vector (for each scalar value of*t*), then what type of value does its derivative**c**′ take?; that is, is d**r**/d*t*=**c**′(*t*) a point, a scalar, a vector, or what?

- If

- Integrating parametrized curves:
- Date assigned: April 2 Monday.
- Date due: April 3 Tuesday.
- Reading from the textbook: Pages 650–654 (§12.2).
- Reading from my notes: Pages 17&18 (the second half of §1.14).
- Exercises due:
If
**r**is a vector-valued function, so that**v**=**r**(*t*) is a vector (for each scalar value of*t*), then:- What type of value
can its definite integrals take?;
that is,
can
∫
^{b}_{t=a}**r**(*t*) d*t*= ∫^{b}_{t=a}**v**d*t*(where*a*and*b*are scalars) be a point, a scalar, a vector, or what? - What type of value
can its indefinite integrals take?;
that is,
can
∫
**r**(*t*) d*t*= ∫**v**d*t*be a point, a scalar, a vector, or what?

- What type of value
can its definite integrals take?;
that is,
can
∫

- Arclength:
- Date assigned: April 3 Tuesday.
- Date due: April 4 Wednesday.
- Reading from the textbook: Pages 656–659 (§12.3).
- Reading from my notes: Page 20 (§1.16).
- Exercises due:
Section 12.3 of the textbook uses several variables,
including
**r**,*s*,*t*,**T**, and**v**, to describe various quantities on the path of a parametrized curve. Fill in the right-hand side of each of these equations with the appropriate one of these variables:- d
**r**/d*t*= ___. **v**/|**v**| = ___.- d
**r**/d*s*= ___.

- d

- Functions of several variables:
- Date assigned: April 5 Thursday.
- Date due: April 9 Monday.
- Reading from the textbook: Pages 676–681 (§13.1).
- Reading from my notes: From page 21 through the top of page 23 (§2.1).
- Exercises due:
- If
*f*(2, 3) = 5, then what number or point must belong to the domain of*f*and what number or point must belong to the range of*f*? - If
*f*(2, 3) = 5, then what point must be on the graph of*f*?

- If

- Limits and continuity:
- Date assigned: April 9 Monday.
- Date due: April 10 Tuesday.
- Reading from the textbook: Pages 684–690 (§13.2).
- Reading from my notes: The rest of page 23 through most of page 25 (§§2.2–2.4).
- Exercises due:
- Suppose that the limit of
*f*approaching (2, 3) is 5 (in symbols, lim_{(x,y)→(2,3)}*f*(*x*,*y*) = 5), and the limit of*g*approaching (2, 3) is 7 (lim_{(x,y)→(2,3)}*g*(*x*,*y*) = 7). What (if anything) is the limit of*f*+*g*approaching (2, 3)? (lim_{(x,y)→(2,3)}(*f*(*x*,*y*) +*g*(*x*,*y*)) = ?). - Suppose that
the limit of
*f*approaching (0, 0) horizontally is 4 (in symbols, lim_{(x,y)→(0,0),y=0}*f*(*x*,*y*) = 4), and the limit of*f*approaching (0, 0) vertically is 6 (lim_{(x,y)→(0,0),x=0}*f*(*x*,*y*) = 6). What (if anything) is the limit of*f*approaching (0, 0)? (lim_{(x,y)→(0,0)}*f*(*x*,*y*) = ?).

- Suppose that the limit of

- Vector fields and differential forms:
- Date assigned: April 10 Tuesday.
- Date due: April 11 Wednesday.
- Reading from my notes: Pages 27&28 (§§3.1–3.3).
- Reading from the textbook: Page 828 and most of page 829 (§15.2, Vector Fields).
- Online notes: Examples of vector fields.
- Exercises due:
- Given
**u**= ⟨*M*,*N*,*O*⟩, express**u**⋅ d**r**as a differential form (using**r**= (*x*,*y*,*z*)). - Given
**v**= ⟨*M*,*N*⟩, express**v**⋅ d**r**and**v**× d**r**as differential forms (using**r**= (*x*,*y*)).

- Given

- Partial derivatives:
- Date assigned: April 11 Wednesday.
- Date due: April 12 Thursday.
- Readings from my notes:
- The rest of page 25 and page 26 (§§2.5&2.6);
- Page 29 and through most of page 31 (§§3.4&3.5).

- Reading from the textbook: Pages 693–702 (§13.3), especially the Examples.
- Exercises due:
- If
*f*is a function of two variables and the partial derivatives of*f*are D_{1}*f*(*x*,*y*) = 2*y*and D_{2}*f*(*x*,*y*) = 2*x*, then what is the differential of*f*(*x*,*y*)? (If you're trying to figure out a formula for the function*f*, then you're doing too much work!) - If
*u*is a variable quantity and the differential of*u*is d*u*=*x*^{2}d*x*+*y*^{3}d*y*, then what are the partial derivatives of*u*with respect to*x*and*y*? (If you're trying to figure out a formula for the quantity*u*, then you're doing too much work!)

- If

- Directional derivatives:
- Date assigned: April 12 Thursday.
- Date due: April 16 Monday.
- Readings from the textbook:
- Read from page 713 through the end of Example 3 on page 717 (all of §13.5 except Gradients and Tangents to Level Curves);
- Read from the bottom of page 718 through page 720 (§13.5: Functions of Three Variables, The Chain Rule for Paths);
- Read the bottom of page 829 and Example 1 on page 830 (§15.2, Gradient Fields).

- Reading from my notes: The rest of page 31 (§§3.6&3.7).
- Exercises due:
Suppose that ∇f(2, 3) = ⟨3/5, 4/5⟩.
- In which direction
**u**is the directional derivative D_{u}*f*(2, 3) the greatest? - In which directions
**u**is the directional derivative D_{u}*f*(2, 3) equal to zero? - In which direction
**u**is the directional derivative D_{u}*f*(2, 3) the least (with a large absolute value but negative)?

- In which direction

- Matrices and the Chain Rule:
- Date assigned: April 17 Tuesday.
- Date due: April 18 Wednesday.
- Reading from the textbook: Pages 704–710 (§13.4).
- Online notes: Matrices as derivatives.
- Exercise due:
If
*u*=*f*(*x*,*y*,*z*) and*v*=*g*(*x*,*y*,*z*), then what is the matrix d(*u*,*v*)/d(*x*,*y*,*z*)?

- Tangents:
- Date assigned: April 18 Wednesday.
- Date due: April 19 Thursday.
- Readings from the textbook:
- The rest of pages 717 and 718 (§13.5, Gradients and tangents to level curves);
- From page 721 through Example 3 on page 723 (§13.6, Tangent planes and normal lines).

- Reading from my notes: The rest of page 33 and the top of page 34 (§3.8).
- Exercises due:
Fill in each blank with ‘line’ or ‘plane’.
- If ∇
*f*(*a*,*b*) exists but is not zero, then*f*has a tanget ___ and a normal ___ through (*a*,*b*). - If ∇
*f*(*a*,*b*,*c*) exists but is not zero, then*f*has a tanget ___ and a normal ___ through (*a*,*b*,*c*).

- If ∇

- Local optimization:
- Date assigned: April 19 Thursday.
- Date due: April 23 Monday.
- Reading from my notes: Pages 37&38 (§3.10).
- Reading from the textbook: Pages 730–736 (§13.7).
- Exercises due:
Consider a function
*f*of two variables that is defined everywhere. Identify whether*f*has a local maximum, a local minimum, a saddle, or none of these at a point (*a*,*b*) with these characteristics:- If the partial derivatives of
*f*at (*a*,*b*) are both negative. - If one of the partial derivatives of
*f*at (*a*,*b*) is zero and the other is negative. - If both partial derivatives of
*f*at (*a*,*b*) are zero and the Hessian determinant of*f*at (*a*,*b*) is negative. - If both partial derivatives of
*f*at (*a*,*b*) are zero, the Hessian determinant of*f*at (*a*,*b*) is positive, and the unmixed second partial derivatives of*f*at (*a*,*b*) are negative. - If both partial derivatives of
*f*at (*a*,*b*) are zero, the Hessian determinant of*f*at (*a*,*b*) is positive, the unmixed second partial derivatives of*f*at (*a*,*b*) are positive, and the mixed second partial derivatives of*f*at (*a*,*b*) are negative.

- If the partial derivatives of

- Constrained optimization:
- Date assigned: April 23 Monday.
- Date due: April 24 Tuesday.
- Reading from the textbook: Pages 739–746 (§13.8).
- Exercise due:
Suppose that you wish to maximize a continuous function
on the region in 3 dimensions
defined in rectangular coordinates
by 0 ≤
*x*≤ 1, 0 ≤*y*≤ 1, and 0 ≤*z*≤ 1. How many different constrained regions will you have to check, and what are they? (Hint: One constrained region to check is the 3-dimensional interior, given by the strict inequalities 0 <*x*< 1, 0 <*y*< 1, and 0 <*z*< 1. There are eight constrained regions given entirely by equations, each of which is a 0-dimensional point:*x*= 0,*y*= 0, and*z*= 0;*x*= 0,*y*= 0, and*z*= 1;*x*= 0,*y*= 1, and*z*= 0;*x*= 0,*y*= 1, and*z*= 1;*x*= 1,*y*= 0, and*z*= 0;*x*= 1,*y*= 0, and*z*= 1;*x*= 1,*y*= 1, and*z*= 0; and*x*= 1,*y*= 1, and*z*= 1. You still need to find the constrained regions of intermediate dimension, each of which will be given partially by strict inequalities and partially by equations. Don't worry too much about the inequalities; what's really necessary is to get the right equation or equations for each constrained region. I strongly suggest that you draw a picture to help keep things straight.)

- Linearization:
- Date assigned: April 25 Wednesday.
- Date due: April 26 Thursday.
- Reading from the textbook: The rest of page 723 and through page 727 (the rest of §13.6).
- Reading from my notes: The rest of page 34 through page 36 (§3.9).
- Exercises due:
- If a function
*f*is to have a good linear approximation in a region, then it's best if its partial derivatives of what order are close to zero in that region? (Its first partial derivatives, its second partial derivatives, its third partial derivatives, or what?) - If (∂
*u*/∂*x*)_{y}= −3 and (∂*u*/∂*y*)_{x}= 2, then is the quantity*u*more or less sensitive to changes in*x*compared to changes in*y*?

- If a function

- Integration along curves:
- Date assigned: April 26 Thursday.
- Date due: April 30 Monday.
- Reading from my notes: Page 39 and all but the very bottom of page 40 (§4.1 and most of §4.2).
- Reading from the textbook:
From page 830 through Example 7 on page 836
(§15.2:
Line Integrals of Vector Fields;
Line Integrals with Respect to
*dx*,*dy*, or*dz*; Work Done by a Force over a Curve in Space; Flow Integrals and Circulation for Velocity Fields). - Exercises due:
- If you wish to integrate the vector field
**F**(*x*,*y*,*z*) = ⟨2*x*, −3*x*, 4*x**y*⟩ = 2*x***i**− 3*x***j**+ 4*x**y***k**along a curve in (*x*,*y*,*z*)-space, then what differential form are you integrating? - If you wish to integrate a vector field
**F**along the circle parametrized by (*x*,*y*) = (sin*t*, cos*t*) for 0 ≤*t*≤ 2π, and if you orient the circle counterclockwise (assuming that the coordinate system is also oriented counterclockwise), then what should be the bounds on your integral in the variable*t*? (That is, is it ∫_{0}^{2π}**F**(sin*t*, cos*t*) · ⟨cos*t*, −sin*t*⟩ d*t*or ∫_{2π}^{0}**F**(sin*t*, cos*t*) · ⟨cos*t*, −sin*t*⟩ d*t*?)

- If you wish to integrate the vector field

- More integration on curves:
- Date assigned: April 30 Monday.
- Date due: May 1 Tuesday.
- Reading from my notes: the very bottom of page 40 and page 41 (the rest of §4.2 and §4.3).
- Readings from the textbook:
- Pages 821–826 (§15.1);
- the rest of page 836 and page 837 (§15.2, Flux Across a Simple Closed Plane Curve).

- Exercises due:
- If you wish to integrate
a scalar field (that is a function of several variables)
on the circle parametrized by
(
*x*,*y*) = (sin*t*, cos*t*) for 0 ≤*t*≤ 2π, then what should be the bounds on your integral in the variable*t*? (That is, is it ∫_{0}^{2π}*f*(sin*t*, cos*t*) d*t*or ∫_{2π}^{0}*f*(sin*t*, cos*t*) d*t*?) - If you wish to integrate the vector field
**F**(*x*,*y*) = ⟨*x*^{2}, 3⟩ =*x*^{2}**i**+3**j**across a curve in the (*x*,*y*)-plane, and if you orient the plane as usual, then what differential form should you integrate along the curve?

- If you wish to integrate
a scalar field (that is a function of several variables)
on the circle parametrized by
(

- Systems of inequalities: Not assigned (lecture only, no homework due).
- Double integrals:
- Date assigned: May 3 Thursday.
- Date due: May 7 Monday.
- Readings from the textbook:
- Page 755 and most of page 756 (§14.1: introduction; Double Integrals);
- Page 757 after Figure 14.3 and through page 759 (§14.1: Fubini's Theorem for Calculating Double Integrals);
- Page 760 and the top half of page 761 (§14.2: introduction; Double Integrals over Bounded, Nonrectangular Regions);
- The theorem on page 762 (Theorem 14.2);
- The paragraph before Example 2 on page 763 and through page 766 (§14.2: Example 2; Finding Limits of Integration; Properties of Double Integrals).

- Reading from my notes: Page 44 and most of page 45 (§5.1).
- Exercises due: Exercises 81&82 from Section 14.2 on page 769.

- Triple integrals:
- Date assigned: May 7 Monday.
- Date due: May 8 Tuesday.
- Readings from the textbook:
- Page 779 and the top of page 780 (§14.5: introduction; Triple Integrals);
- From the bottom of page 780 to the end of Example 3 on page 784
(§14.5:
Finding Limits of Integration in the Order
*dz**dy**dx*); - The middle of page 785 (§14.5: Properties of Triple Integrals).

- Exercise due: In how many ways can you order 3 variables of integration? List them.

- Areas, volumes, and averages:
- Date assigned: May 8 Tuesday.
- Date due: May 9 Wednesday.
- Readings from the textbook:
- The bottom of page 756 and the top of page 757 (§14.1: Double Integrals as Volumes);
- The rest of pages 761–763 (§14.2: the rest of Volumes);
- Pages 769–772 (§14.3);
- The rest of page 780 (§14.5: Volume of a Region in Space);
- The rest of pages 784&785 (§14.5: Average value of a function in space).

- Exercises due:
Suppose that
*a*<*b*and*c*<*d*are four real numbers, that*f*is a continuous function of two variables whose domain is the rectangle {*x*,*y*|*a*≤*x*≤*b*,*c*≤*y*≤*d*}, and that*f*takes only positive values. Write down expressions (in terms of*a*,*b*,*c*,*d*, and*f*) for the volume under the graph of*f*:- Using ideas from §14.2,
as an iterated double integral in the variables
*x*and*y*; - Using ideas from §14.5,
as an iterated triple integral
in the variables
*x*,*y*, and*z*.

*f*is; however, you should be able to begin evaluating your answer to #2 if you write the variables in an appropriate order, and this should turn it into your answer from #1, after which you shouldn't be able to go any further.) - Using ideas from §14.2,
as an iterated double integral in the variables

- Moments:
- Date assigned: May 9 Wednesday.
- Date due: May 10 Thursday.
- Reading from the textbook: Pages 788–793 (§14.6).
- Exercises due:
- Give the formulas for the centre of mass
(
*x̄*,*ȳ*,*z̄*) of a three-dimensional solid in terms of the total mass*M*and the moments*M*_{x,y},*M*_{x,z}and*M*_{y,z}. - Give a formula for the polar moment of inertia
*I*_{0}of a two-dimensional plate in terms of the moments of inertia*I*_{x}and*I*_{y}about the coordinate axes.

- Give the formulas for the centre of mass
(

- Polar coordinates:
- Date assigned: May 10 Thursday.
- Date due: May 14 Monday.
- Reading from my notes: From the middle of page 50 through page 51 (most of §5.5).
- Readings from the textbook:
- Pages 773–777 (§14.4);
- Pages 795–802 (§14.7).

- Exercises due:
- Give a formula for the area element in the plane
in rectangular coordinates
*x*and*y*. (Answer: d*x*d*y*, or more properly |d*x*∧ d*y*|; either is acceptable, as are d*y*d*x*and |d*y*∧ d*x*|.) - Give a formula for the area element in the plane
in polar coordinates
*r*and*θ*. - Give a formula for the volume element in space
in rectangular coordinates
*x*,*y*, and*z*. (Answer: d*x*d*y*d*z*, or more properly |d*x*∧ d*y*∧ d*z*|; either is acceptable, as are the the other orders.) - Give a formula for the volume element in space
in cylindrical coordinates
*r*,*θ*, and*z*. - Give a formula for the volume element in space
in spherical coordinates
*ρ*,*θ*, and*φ*.

- Give a formula for the area element in the plane
in rectangular coordinates

- Coordinate transformations:
- Date assigned: May 15 Tuesday.
- Date due: May 16 Wednesday.
- Reading from my notes: The rest of pages 45–50 (§§5.2–5.4 and the rest of §5.5).
- Reading from the textbook: Pages 806–814 (§14.8).
- Exercise due:
If
*x*=*f*(*u*,*v*) and*y*=*g*(*u*,*v*), where*f*and*g*are continuously differentiable everywhere, then write the area element d*x*d*y*(which is more properly written |d*x*∧ d*y*|) in terms of*u*,*v*, their differentials, and the partial derivatives of*f*and*g*(which you can also think of as the partial derivatives of*x*and*y*with respect to*u*and*v*). (There are formulas in both the notes and the textbook that you can use, or you can work it out from first principles using the more proper expression involving d*x*and d*y*given above.)

- Surfaces:
- Date assigned: May 16 Wednesday.
- Date due: May 17 Thursday.
- Readings from the textbook:
- Pages 632–635 (§11.6);
- Page 863 and through Example 3 page 865 (§15.5: introduction; Parametrizations of Surfaces).

- Reading from my notes: Page 52 and the top half of page 53 (§§6.1&6.2).
- Exercises due:
- Write down a parametrization of the sphere
*x*^{2}+*y*^{2}+*z*^{2}= 1 using the spherical coordinates*φ*and*θ*. - Write down a parametrization of the portion of the cone
*x*^{2}+*y*^{2}=*z*^{2}where 0 ≤*z*≤ 1 using cylindrical coordinates (either*z*and*θ*or*r*and*θ*).

- Write down a parametrization of the sphere

- Integrals across surfaces:
- Date assigned: May 17 Thursday.
- Date due: May 21 Monday.
- Reading from the textbook: Most of page 878 and through Example 6 on page 881 (§15.6: Orientation of a Surface; Surface Integrals of Vector Fields).
- Reading from my notes: The rest of page 53 and through most of page 55 (§§6.3–6.5).
- Exercises due:
- If you parametrize a closed surface containing the origin
using the spherical coordinates
*φ*and*θ*(using the U.S. mathematicians' convention for which of these is which) and orient (by which I technically mean pseudoorient) this surface outwards, then (using the right-hand rule to interpret this as an honest orientation) does this orientation correspond to d*φ*∧ d*θ*(that is increasing*φ*followed by increasing*θ*) or to d*θ*∧ d*φ*(that is increasing*θ*followed by increasing*φ*)? - Write down a formula for the pseudooriented surface element
d
**S**=**n**d*σ*on a parametrized surface in terms of the coordinates (*x*,*y*,*z*) and/or the parameters (*u*,*v*) and their differentials and/or partial derivatives. (There are multiple correct answers to this throughout the readings from the textbook and the handout.)

- If you parametrize a closed surface containing the origin
using the spherical coordinates

- Integrals on surfaces:
- Date assigned: May 21 Monday.
- Date due: May 22 Tuesday.
- Reading from my notes: The rest of page 55 and page 56 (§6.6).
- Readings from the textbook:
- The rest of page 865 and through page 871 (§15.5: Surface Area; Implicit Surfaces);
- From page 874 through the end of Example 4 on page 878 (§15.6: introduction; Surface Integrals);
- The rest of page 881 through page 883 (§15.6: Moments and Masses of Thin Shells).

- Exercise due:
Write down a formula for the surface area element
d
*σ*= |d**S**| on a parametrized surface in terms of the coordinates (*x*,*y*,*z*) and/or the parameters (*u*,*v*) and their differentials and/or partial derivatives. (There are multiple correct answers to this throughout the readings from the textbook and the handout.)

- Conservative vector fields and exact differential forms:
- Date assigned: May 23 Wednesday.
- Date due: May 24 Thursday.
- Reading from my notes: Pages 42&43 (§4.4).
- Reading from the textbook: Pages 840–849 (§15.3).
- Exercises due (true or false):
- If
*f*is a differentiable scalar field (a function of several variables), then its gradient, the vector field ∇*f*, must be conservative. - If
*u*is a differentiable scalar quantity, then its differential, the differential form d*u*, must be exact. - If
**F**is a conservative vector field in two dimensions, then the differential form**F**(*x*,*y*) ⋅ d(*x*,*y*) must be exact. - If
**F**is a vector field in two dimensions and the differential form**F**(*x*,*y*) ⋅ d(*x*,*y*) is exact, then**F**must be conservative.

- If

- Green's Theorem:
- Date assigned: May 24 Thursday.
- Date due: May 29 Tuesday.
- Readings from my notes:
- From page 57 through the top of page 59 (§7.1);
- Optional: §7.2;
- From page 60 through the top of page 62 (§7.3).

- Reading from the textbook: Pages 851–861 (§15.4).
- Exercise due: Write down as many different versions of the general statement of Green's Theorem as you can think of. (There are some in both the textbook and the handout. I'll give full credit for at least two that are different beyond a trivial change in notation, but there are really more than that.)

- Stokes's Theorem:
- Date assigned: May 29 Tuesday.
- Date due: May 30 Wednesday.
- Reading from my notes: The rest of page 62 and the top of page 63 (§7.4).
- Reading from the textbook: Pages 885–895 (§15.7).
- Exercises due:
- Suppose that you have a compact surface in 3-dimensional space,
the
*z*-axis passes through this surface, the surface is oriented (by which I really mean pseudooriented) so that*z*is increasing along the*z*-axis through the surface, and you orient the boundary of this surface using the right-hand rule as usual. Is the cylindrical coordinate*θ*increasing or decreasing overall along the boundary curve? - Given
*f*(*x*,*y*,*z*) = 2*x*^{3}*y*^{2}cos(e^{sin(z)}), what is ∇ × ∇*f*, the curl of the gradient of*f*? (Hint: If you're doing a bunch of calculations, then you're making this too hard.)

- Suppose that you have a compact surface in 3-dimensional space,
the

- Gauss's Theorem:
- Date assigned: May 30 Wednesday.
- Date due: May 31 Thursday.
- Reading from my notes: The rest of page 63 (§7.5).
- Reading from the textbook: Pages 897–906 (§15.8).
- Exercises due:
- Suppose that you have a compact region in 3-dimensional space,
the origin lies within this region,
and you orient (by which I really mean pseudoorient) the boundary as usual.
Is the spherical coordinate
*ρ*increasing or decreasing overall through the boundary surface? - Given
**F**(*x*,*y*,*z*) = ⟨2*x*^{3}*y*^{2}, cos(e^{sin(z)}), sin(e^{cos(z)})⟩ = 2*x*^{3}*y*^{2}**i**+ cos(e^{sin(z)})**j**+ sin(e^{cos(z)})**k**, what is ∇ ⋅ ∇ ×**F**, the divergence of the curl of**F**? (Hint: If you're doing a bunch of calculations, then you're making this too hard.)

- Suppose that you have a compact region in 3-dimensional space,
the origin lies within this region,
and you orient (by which I really mean pseudoorient) the boundary as usual.
Is the spherical coordinate

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