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Math 20A lecture 1
An overview, some administration, and a littlegeometry
T.J. Barnet-Lamb
Brandeis University
Math 20A lecture 1 – p. 1/13
Administration
Instructor Thomas Barnet-Lamb, Goldsmith [email protected]
Schedule Tues, Fri 10.30am–12 noonOffice hours Tues 2–3.30pm, Fri 3–4.30pmTextbook Multivariable Calculus, Concepts and
Contexts, James Stewart, 4th ed
Math 20A lecture 1 – p. 2/13
What will we do in Math 20A
Math 20A lecture 1 – p. 3/13
What will we do in Math 20A
1. Calculus of Several Variables
Math 20A lecture 1 – p. 3/13
What will we do in Math 20A
1. Calculus of Several Variables
2. Analytic Geometry of space
Math 20A lecture 1 – p. 3/13
What will we do in Math 20A
1. Calculus of Several Variables
2. Analytic Geometry of space
Math 20A lecture 1 – p. 3/13
How to get an A
Your grade will be composed of:35% Weekly homeworks
(set on Fri, due following Fri)30% Midterm (Oct 2nd)35% Final exam
Math 20A lecture 1 – p. 4/13
Why math?
I think:
Math 20A lecture 1 – p. 5/13
Why math?
I think:
The objective of mathematical enquiry is tostudy abstract objects—objects which exist only inthought—unravelling their properties with thefollowing end in mind: that when we encounterreal-world objects which are analogous, we canapply by analogy the knowledge we have gained.
Math 20A lecture 1 – p. 5/13
Guess where?
Math 20A lecture 1 – p. 6/13
A true story
Math 20A lecture 1 – p. 7/13
A true story
Math 20A lecture 1 – p. 8/13
The story so far
Relative positions are an example of a vectorquantity: one which has a direction as well as magnitude.
If we want to process these relative positionsmathematically, we often have to turn them in to numbers,but this can only be done using a frame of reference, whichis usually arbitrary.
Most things you can do with these numbers will benonsensical because they’ll depend on the frame ofreference in a silly way.
But there are some things we can do: take the sum,negative, and length of a vector, or multiply a vector by anordinary number.
Math 20A lecture 1 – p. 9/13
Does the universe have a center?
How many Harvard Students does it take to change alightbulb?
Math 20A lecture 1 – p. 10/13
Does the universe have a center?
How many Harvard Students does it take to change alightbulb?
Just one: they hold it in the socket and wait for the world torevolve around them.
Math 20A lecture 1 – p. 10/13
Example: manipulating vectorsgeometrically
Consider the following vectors, u and v. (See picture onboard.)
Draw u + v
Draw u − v
Math 20A lecture 1 – p. 11/13
Example: manipulating vectors withnumbers
Let u = 〈1, 2, 3〉 and v = i − j.Compute |v|, 2u, v + u and |v + 2u|.
Math 20A lecture 1 – p. 12/13
Example: manipulating vectors withnumbers
Let u = 〈1, 2, 3〉 and v = i − j.Compute |v|, 2u, v + u and |v + 2u|.Well, |v| =
√
12 + (−1)2 =√
2
Math 20A lecture 1 – p. 12/13
Example: manipulating vectors withnumbers
Let u = 〈1, 2, 3〉 and v = i − j.Compute |v|, 2u, v + u and |v + 2u|.Well, |v| =
√
12 + (−1)2 =√
2, 2u = 〈2, 4, 6〉
Math 20A lecture 1 – p. 12/13
Example: manipulating vectors withnumbers
Let u = 〈1, 2, 3〉 and v = i − j.Compute |v|, 2u, v + u and |v + 2u|.Well, |v| =
√
12 + (−1)2 =√
2, 2u = 〈2, 4, 6〉, andv + u = 〈2, 1, 3〉.
Math 20A lecture 1 – p. 12/13
Example: manipulating vectors withnumbers
Let u = 〈1, 2, 3〉 and v = i − j.Compute |v|, 2u, v + u and |v + 2u|.Well, |v| =
√
12 + (−1)2 =√
2, 2u = 〈2, 4, 6〉, andv + u = 〈2, 1, 3〉. Finally, v + 2u = 〈3, 3, 6〉, which haslength
√32 + 32 + 62 =
√9 + 9 + 36 =
√54.
Math 20A lecture 1 – p. 12/13
Example: manipulating vectors withnumbers
Let u = 〈1, 2, 3〉 and v = i − j.Compute |v|, 2u, v + u and |v + 2u|.Well, |v| =
√
12 + (−1)2 =√
2, 2u = 〈2, 4, 6〉, andv + u = 〈2, 1, 3〉. Finally, v + 2u = 〈3, 3, 6〉, which haslength
√32 + 32 + 62 =
√9 + 9 + 36 =
√54.
Find a unit vector in the same direction as 〈2, 3, 6〉.
Math 20A lecture 1 – p. 12/13
Example: manipulating vectors withnumbers
Let u = 〈1, 2, 3〉 and v = i − j.Compute |v|, 2u, v + u and |v + 2u|.Well, |v| =
√
12 + (−1)2 =√
2, 2u = 〈2, 4, 6〉, andv + u = 〈2, 1, 3〉. Finally, v + 2u = 〈3, 3, 6〉, which haslength
√32 + 32 + 62 =
√9 + 9 + 36 =
√54.
Find a unit vector in the same direction as 〈2, 3, 6〉.The length of this vector is√
22 + 32 + 62 =√
4 + 9 + 36 =√
49 = 7. Thus if wemultiply this vector by 1
7, the resulting vector will have
unit length. (And we will not have changed thedirection.) So the answer is 〈2
7, 3
7, 6
7〉
Math 20A lecture 1 – p. 12/13
Example: coordinates
Find the distance from the point (1, 2, 3) tothe xy planethe x axis
The triangle PQR, where
P = (1, 1, 1), Q = (3, 3, 3), R = (−1,−1, 2)
is isosceles: which two sides have the same length?
Math 20A lecture 1 – p. 13/13