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WORKSHEETS for MATH 233H Multivariable Calculus
Instructor: Michael Taylor
Contents
- Tu, 08/11, xx1.1{1.3, Basic one variable calculus (review)
- Th, 08/13, x2.1, Euclidean spaces
- Tu, 08/18, x2.2, Vector spaces and linear transformations
- Th, 08/20, x2.3, Determinants A. Supplementary worksheet { more on determinants
- Th, 08/27, x3.1, Curves and arclength
- Tu, 09/01, x3.2, Exponential and trigonometric functions
- Th, 09/03, x3.2, Exponential and trigonometric functions, II B. Supplementary worksheet { making a trig table
- Tu, 09/08, x3.3, Curvature of planar curves
- Th, 09/10, x3.4, Curvature and torsion of curves in R^3 C. Supplementary worksheet { the matrix exponential
- Tu, 09/15, Review for test Th, 09/17, Test
- Tu, 09/22, x4.1, The derivative in several variables
- Th, 09/24, xx4.1{4.2, The derivative in several variables II
- Tu, 09/29, x4.2, Higher derivatives and power series
- Th, 10/01, x4.2, Higher derivatives II, critical points
- Tu, 10/06, x4.3, Inverse function theorem
- Th, 10/08, x4.3, Implicit function theorem
- Tu, 10/13, x1.2, The Riemann integral in one variable
- Th, 10/15, x5.1, The Riemann integral in n variables
- Tu, 10/20, Review for test Th, 10/22, Test
- Tu, 10/27, x5.1, The Riemann integral in n variables II, iterated integrals
- Th, 10/29, x5.1, The Riemann integral in n variables III, change of vari- able formulas
- Tu, 11/03, x6.1, Surfaces and surface integrals
- Th, 11/05, x6.1, Surfaces and surface integrals II D. Supplementary worksheet { partitions of unity
- Tu, 11/10, x6.3, Formulas of Gauss, Green, and Stokes
- Th, 11/12, x6.3, Formulas of Gauss, Green, and Stokes II
- Tu, 11/17, Review of course (last day of class)
Introduction
These worksheets serve to guide the student through the text for Math 233H, Multivariable Calculus, by M. Taylor. They are designed so that each worksheet covers the material of one lecture. Each worksheet deals with material in a desig- nated section of the text, and the idea is that a student can do the exercises in a worksheet in consultation with the text, and in that manner master the material in the text. There are also a handful of supplementary worksheets, to compensate for time lost due to the transition from in class to remote instruction. These worksheets have been produced in response to the health crisis of 2020. They are dated to correspond to a class meeting twice a week.
Worksheet 2, Thursday, 08/
x2.1, Euclidean spaces
- De ne the vector operations on Rn.
- Given x; y 2 Rn, de ne the dot product x y.
- Given x 2 Rn, we de ne the norm jxj 2 [0; 1 ) by
jxj =
p x x:
Consult Proposition 2.1.1 and show that the triangle inequality
jx + yj jxj + jyj
follows from Cauchy's inequality
jx yj jxj jyj:
- Consult Proposition 2.1.2 for the proof of Cauchy's inequality.
- Given pj 2 Rn, de ne what it means to say
pj converges to p as j! 1. (pj ) is Cauchy.
- Given S Rn, de ne what it means to say
S is closed; S is open.
- Given x; y 2 Rn, we say
x? y () x y = 0:
Show that jx + yj^2 = jxj^2 + jyj^2 () x? y:
Worksheet 3, Tuesday, 08/
x2.2, Vector spaces and linear transformations
- De ne the concept of a vector space (over F = R or C). Note that Rn^ is a vector space over R and Cn^ is a vector space over C. (Fn^ is a vector space over F.)
- Let S = fv 1 ; : : : ; vkg V , a vector space. De ne what it means to say
S spans V , S is linearly independent, S is a basis of V.
- Study Lemma 2.2.1 and Proposition 2.2.2, whose content is that If V has a basis fv 1 ; : : : ; vkg and if fw 1 ; : : : ; wℓg V is linearly independent, then ℓ k. Show that this leads to Corollary 2.2.3: If V is nite dimensional, then any two bases of V have the same number of elements. In such a case, dim V denotes the number of elements in a basis of V.
- State Propositions 2.2.4 and 2.2.5.
- State Proposition 2.2.6, the Fundamental Theorem of Linear Algebra, and show how this follows from Propositions 2.2.4 and 2.2.5.
- Deduce from the Fundamental Theorem of Linear Algebra that if V is nite dimensional and A : V! V is linear, then
A injective , A surjective , A isomorphism:
- State Proposition 2.2.9, characterizing when a matrix A 2 M (n; F) is invertible, in terms of the behavior of its columns.
Supplementary worksheet x2.3, more on determinants
- Verify the following method of computing 3 3 determinants. Given
A =
a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33
A ;
form a 3 5 rectangular matrix by copying the rst two columns of A to the right. The products in (2.3.16) with plus signs are the products of each of the three downward sloping diagonals marked in bold below.
0
@
a 11 a 12 a 13 a 11 a 12 a 21 a 22 a 23 a 21 a 22 a 31 a 32 a 33 a 31 a 32
A :
The products in (2.3.16) with minus signs are the products of each of the three upward sloping diagonals marked in bold below.
0
@
a 11 a 12 a 13 a 11 a 12 a 21 a 22 a 23 a 21 a 22 a 31 a 32 a 33 a 31 a 32
A :
- Use the method described above to compute the determinants of
0 @
A ;
A ;
A :
- Given A = (0 1 2), compute det AtA and det AAt.
- Compute the determinant of
0 B @
C
A :
Worksheet 5, Thursday, 08/
x3.1, Curves and arclength
- Let : [a; b]! Rn^ be a C^1 curve, with
velocity ′(t) and speed j ′(t)j:
Study the statement and proof of Proposition 3.1.1, leading to the formula
∫ (^) b
a
j ′(t)j dt
for the length of this curve.
- State what it means for : [ ; ]! Rn^ to be a reparametrization of , and study the identity ℓ() = ℓ( ); in (3.1.17).
- Discuss the reparametrization by arclength of , given ′^ nowhere vanishing.
- Read about the parametrization of the unit circle S^1 by arclength, discussed in (3.1.22){(3.1.34), leading to the de nition of the trigonometric functions
cos t; sin t:
- Discuss the derivation in (3.1.34){(3.1.39) of the identities
d dt
cos t = sin t;
d dt
sin t = cos t:
- Work out the arclength calculations proposed in Exercises 1{4 at the end of x3.1.
- Review (3.2.29){(3.2.37), to the effect that
(t) = eit; t 2 R
is a unit-speed parametrization of the unit circle.
- Recall de nitions from basic trigonometry (see Exercise 4 of Worksheet 5) and deduce from Exercise 6 above that
eit^ = cot t + i sin t; t 2 R:
This is called Euler's identity.
- Deduce from Exercise 7 that
d dt
eit^ = ieit^ =)
d dt
cos t = sin t;
d dt
sin t = cos t:
- Deduce from Exercise 4 that
ei(s+t)^ = eiseit:
Combine this with Exercise 7 to derive formulas for
cos(s + t); sin(s + t):
Worksheet 7, Thursday, 09/
x3.2, Exponential and trigonometric functions, II
- De ne
tan t =
sin t cos t
; sec t =
cos t
and check out Exercises 3{5 at the end of x3.2 of the text.
- Check out Exercises 6{7 at the end of x3.2, leading to the formula
6
0
dx p 1 x^2
- Look at Exercises 2 and 5 at the end of x3.2, and show that
6
∫ p 3 = 3
0
dx 1 + x^2
- Set
cosh t =
(et^ + e t); sinh t =
(tt^ e t);
and show that d dt
cosh t = sinh t;
d dt
sinh t = cosh t;
cosh^2 t sinh^2 t = 1:
- Check out Exercise 14 at the end of x3.2. This involves evaluating
I(u) =
∫ (^) u
0
dv p 1 + v^2
in two ways: v = sinh y; v = tan t:
Show how this leads to the identity
∫ (^) x
0
sec t dt = sinh ^1 (tan x); for jxj <
- Check out the study of the functions xr^ , de ned by
xr^ = er^ log^ x; x > 0 ; r 2 C;
in Exercises 18{22 at the end of x3.2. Show that
d dx
xr^ = rxr ^1 :
Supplementary worksheet
x3.2, Making a trig table
This worksheet is for students who have access to computer software allowing for numerical calculation (such as Matlab).
Follow exercises 31{39 at the end of x3.2, and make a table of
(1) cos ℓ◦^ and sin ℓ◦;
for the integers ℓ between 0 and 45. Here
(2) ℓ◦^ =
radians:
A basis for the calculation of (1) is given by the identities
ei=^3 =
(1 + i
p 3);
ei=^4 =
p 2
(1 + i);
e^2 i=^5 = c 5 + is 5 ;
where
(4) c 5 =
p 5 1); s 5 =
1 c^25 ;
related to regular n-gons (n = 6; 4 ; 5), which are established in the text. Note that
3
and using the identity e(^ ^ )i^ = e ie ^ i
repeatedly gives (1) when ℓ is an integral multiple of 3.
Exercise 39 presents a cube root construction that allows one to handle ℓ = 1.
Exercise 33 discusses numerical evaluation of square roots, such as arise in (3){(4).
Worksheet 8, Tuesday, 09/
x3.3, Curvature of planar curves
- Consult (3.3.1){(3.3.8) to see that, if : (a; b)! R^2 is a unit speed planar curve, then its unit tangent vector T (s) and normal N (s) at (s) are given by
′(s) = T (s); N (s) = JT (s);
where
J =
is counterclockwise rotation by 90◦.
Then the curvature (s) at (s) is given by
T ′(s) = (s)N (s); i.e., T ′(s) = (s)JT (s):
- Consider the treatment in (3.3.9){(3.3.29) of the problem of constructing a unit speed curve (s) with given curvature (s). Note the use of the matrix exponential, de ned in (3.3.18) and used in (3.3.19) and in (3.3.29). See Auxiliary worksheet C for material on the matrix exponential.
- Now suppose we have a smooth curve c : ( ; )! R^2 , not necessarily unit speed, with velocity and acceleration
v(t) = c′(t); a(t) = v′(t);
so
T (t) =
v(t) ∥v(t)∥
; N (t) = JT (t):
Consider Exercises 1{2 at the end of x3.3, expressing the acceleration a(t) as a linear combination of T (t) and N (t):
a(t) =
d^2 s dt^2
T (t) + (t)
( (^) ds
dt
N (t);
and deriving the following formula for the curvature (t) in this situation:
(t) =
a(t) Jv(t) ∥v(t)∥^3
- Find the curvatures of the curves given in Exercises 3{6 at the end of x3.3.
- Consult (3.4.20){(3.4.30) regarding the following problem: Given smooth functions (s) ad (s), nd a unit speed curve (s) for which the solution (T; N; B) to the Frenet-Serret equations is the Frenet frame. Consult (3.4.31){(3.4.44) for a treatment of the special case where and are constant. One again sees the matrix exponential, in (3.4.44). (Compare Exercise 2 in Worksheet 8.)
- Consult Exercises 1{3 at the end of x3.4, expressing the acceleration a(t) = v′(t) as a linear combination of T (t) and N (t),
a(t) =
d^2 s dt^2
T (t) + (t)
( (^) ds dt
N (t)
(compare Exercise 3 of Worksheet 9), and deriving the formula
B =
v a ∥v∥^3
for the curvature and binormal B, and furthermore deriving the formula
(v a) a′ ∥v a∥^2
for the torsion.
- Compute the Frenet frame and the curvature and torsion for the curve c(t) given in Exercise 4 at the end of x3.4.
Supplementary worksheet
xC.4, The matrix exponential
- As treated in Appendix C.4, the matrix exponential is de ned by
etA^ =
∑^1
k=
tk k!
Ak; t 2 R; A 2 M (n; C):
Consult Exercise 1 at the end of xC.4 for a discussion of convergence issues.
- Show that term by term differentiation of the power series given above for etA gives d dt
etA^ = AetA^ = etAA:
- Show that d dt
(etAe tA) = 0;
and hence (etA) ^1 = e tA:
- Show that d dt
(e(s+t)Ae tA) = 0;
and hence that
e(s+t)A^ = esAetA; 8 s; t 2 R; A 2 M (n; C):
- Show that, given A; B 2 M (n; C), if A and B commute, i.e., if
AB = BA;
then, for t 2 R, et(A+B)^ = etAetB^ :
Hint. To start, show that commutativity yields
d dt
(et(A+B)e tB^ e tA) = 0:
- Show that the chain rule plus Exercise 2 above give
d dt
eφ(t)A^ = φ′(t)Aeφ(t)A:
when φ : R! R is a differentiable function.
5A. Review how the trig functions cos t and sin t arise via taking the unit circle centered at the origin in R^2 and parametrizing it by arc length, C(t) = (cos t; sin t). Review how
d dt
C(t) C(t) 0 =)
d dt
cos t = sin t;
d dt
sin t = cos t:
- Review x3.2, on the exponential and trigonometric functions, with attention to power series formula for ez^ , fact that (d=dt)eat^ = aeat, for t 2 R; a 2 C, identity e(a+b)t^ = eatebt; t 2 R; a; b 2 C, log x as the inverse function to x = et; t 2 R; x 2 (0; 1 ).
Review the derivation of Euler's formula,
eit^ = cos t + i sin t;
from the fact that (t) = eit^ is a unit-speed parametrization of the unit circle. See how (d=dt)eit^ = ieit^ leads to formulas for the derivatives of cos t and sin t, rederiving such formulas stated in Exercise 5A above.
- Continue the review of x3.2, with attention to tan t; sec t, and their derivatives, , integral formulas and numerical approximation of this number, cosh t; sinh t, and their derivatives, xr^ = er^ log^ x, and its derivative, two approaches to the evaluation of the length of a parabolic arc, i.e., of ∫ (^) x
0
1 + t^2 dt:
- Review x3.3, on the curvature of a planar curve , with attention to unit tangent T and normal N = JT , curvature , given by dT =ds = N , where s is the arc length parameter, formula for when is not parametrized by arc length, use of the matrix exponential (see xC.4) in solving dT ds
= (s)JT (s):
- Review x3.4, on the curvature and torsion of a curve in R^3 , with attention to velocity v(t) = ′(t) and unit tangent T (t) = v(t)=∥v(t)∥, curvature (s) = ∥dT =ds∥, and normal N , satisfying dT =ds = (s)N (s), binormal B(s) = T (s) N (s), Frenet-Serret formulas dT ds
= N;
dN ds
= T + B;
dB ds
= N;
formulas for ; B, and when is not parametrized by arc length.
- Computations. Worksheets 1{9 point to various computational exercises in the text. Review these.
