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# MAT 237 Formulas

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University of Toronto St. George

Mathematics

MAT237Y1

Robert Brym

Fall

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MAT 237Y la doc
The Geometry of Euclidean Space
VECTORS IN R"
A vector in R" can be written as u
Definitions
Some Properties
u e R
ER
ue R
Definition
u (r......u.) The norm is
Properties
TT ab
volume ellipsoid
haoc
Ical-lalul.
Page 1 ot 23
DOT PRODUCT
Properties
4) u u 20. moreover, u u u -0
More Properties
2) lu l'1 (Cauchy-Schwartz)
The Cosine Law
Theorem
Some Basic Consequences
2) u v 0 means
u-v 0 means 2.
u-v <0 means
PROJECTION
Page 20123
MAT 237Y la doc The Geometry of Euclidean Space VECTORS IN R" A vector in R" can be written as u Definitions Some Properties u e R ER ue R Definition u (r......u.) The norm is Properties TT ab volume ellipsoid haoc Ical-lalul. Page 1 ot 23 DOT PRODUCT Properties 4) u u 20. moreover, u u u -0 More Properties 2) lu l'1 (Cauchy-Schwartz) The Cosine Law Theorem Some Basic Consequences 2) u v 0 means u-v 0 means 2. u-v <0 means PROJECTION Page 20123MAT237 Yla doc
Geometry
AB x AC
Theorem
proju v u
LINES IN R
(r, y
some ways to characterize the line that passes through a piven poirt Fa-(ro.yo.:o) and fllows the
v (a.b.c). If P (r,y. denotes any point on the line. then:
direction of anon-zerovector
PaP 2v uvector equation).
zo (parametric equations)
(symmetric equations).
THE C
PRODUCT
Definition
producIofu and v.
Page 3 of 23
MAT237Y1a doce
Note: uxv det ui 2 i
Some Basis Properties
2) uxu 0 More gereral, if ull v.then uxv 0.
4) ux (v +w) ux v uxw
Note: The cross product is not associative.
The Geometry of the Cross Product
1) If u 0 and v 0. then lux vl Iullvlsin e
Theorem
u (uxv)-0 and v (ux 0. Geometrically, (uxv)iu. (ux v Lv.
Equations of Planes in R3
An equation for the plane that passes through Pa -(ro. yo.%) and is perpendicular (normal to (a, b,c)
Another way to describe a plane is y yo +su, +n, .This is he parametric equation through (ro.yo.zo)
and parallel to u and v.
Theorem
Let u 40. v 0. w A0.
The area of the parallelogram generated by u andvis luxvl
2) The volume of the parallelepiped generated by the vectors u, v. w is lu (vxw).
COORDINATES IN R3
MAT237 Yla doc Geometry AB x AC Theorem proju v u LINES IN R (r, y some ways to characterize the line that passes through a piven poirt Fa-(ro.yo.:o) and fllows the v (a.b.c). If P (r,y. denotes any point on the line. then: direction of anon-zerovector PaP 2v uvector equation). zo (parametric equations) (symmetric equations). THE C PRODUCT Definition producIofu and v. Page 3 of 23 MAT237Y1a doce Note: uxv det ui 2 i Some Basis Properties 2) uxu 0 More gereral, if ull v.then uxv 0. 4) ux (v +w) ux v uxw Note: The cross product is not associative. The Geometry of the Cross Product 1) If u 0 and v 0. then lux vl Iullvlsin e Theorem u (uxv)-0 and v (ux 0. Geometrically, (uxv)iu. (ux v Lv. Equations of Planes in R3 An equation for the plane that passes through Pa -(ro. yo.%) and is perpendicular (normal to (a, b,c) Another way to describe a plane is y yo +su, +n, .This is he parametric equation through (ro.yo.zo) and parallel to u and v. Theorem Let u 40. v 0. w A0. The area of the parallelogram generated by u andvis luxvl 2) The volume of the parallelepiped generated by the vectors u, v. w is lu (vxw). COORDINATES IN R3MAT237 Y1a doc
Rectangular
Cylindrical
Example
Rectangular (r, y, z)
Cylindrical (r,0. z)
Spherical (p. e, d)
2,0,2
Equations of Transformations
Rectangular and Cylindrical: y ar sin 0 and 6- arctan
esin
Rectangular and spherical y-psinesino
VECTORS IN RW
Definition
Definition
Let P (pi..... pn) and ea (a,.....q.) be two points in R Then the dislance between P and ois
Definition
Page of 23
MAT237 Y1a/doc
a.), then the line that
direction of the vector v is the set of all points (ri,... such that
Problem
Suppose that the real numbers a. .e.dsatisly the condition a +(b-2) (c+1) (d-3) -16. Find the
values fora, b. c, d for which sla, b. c. d) (a-7) (b-s): (c-1o)' (d-12) take its maximum and
In this case, we can view a -20
+(c+1) (d-3)' 16 as a "splere centered at (0,2-13)
with radius 4, and fla.b.c d) as the distance from (a, b.c.d) to (7,8 8.-10.
Now the line that joins the center of the "sphere" (o.2.-13) with the point (7.8.-10.12) is given by
these two points (which we can solve easily will give the maximum value for f(a,b,
and the
other one will give the minimum.
Limits
Definition: Limit of One Variable
When we say The function S() approaches the number L as rapproaches a" (write lim L). what we
Geometrically: For any open interval B that contains we can always find an open interval A that
contains a such that tor all xe A, a f(x)e B.
Algebraically: For every e 0. there exists 0 such that o 0, there exists 0 such that
Page 6 of 23
MAT237 Y1a doc Rectangular Cylindrical Example Rectangular (r, y, z) Cylindrical (r,0. z) Spherical (p. e, d) 2,0,2 Equations of Transformations Rectangular and Cylindrical: y ar sin 0 and 6- arctan esin Rectangular and spherical y-psinesino VECTORS IN RW Definition Definition Let P (pi..... pn) and ea (a,.....q.) be two points in R Then the dislance between P and ois Definition Page of 23 MAT237 Y1a/doc a.), then the line that direction of the vector v is the set of all points (ri,... such that Problem Suppose that the real numbers a. .e.dsatisly the condition a +(b-2) (c+1) (d-3) -16. Find the values fora, b. c, d for which sla, b. c. d) (a-7) (b-s): (c-1o)' (d-12) take its maximum and In this case, we can view a -20 +(c+1) (d-3)' 16 as a "splere centered at (0,2-13) with radius 4, and fla.b.c d) as the distance from (a, b.c.d) to (7,8 8.-10. Now the line that joins the center of the "sphere" (o.2.-13) with the point (7.8.-10.12) is given by these two points (which we can solve easily will give the maximum value for f(a,b, and the other one will give the minimum. Limits Definition: Limit of One Variable When we say The function S() approaches the number L as rapproaches a" (write lim L). what we Geometrically: For any open interval B that contains we can always find an open interval A that contains a such that tor all xe A, a f(x)e B. Algebraically: For every e 0. there exists 0 such that o 0, there exists 0 such that Page 6 of 23MAT237 Y1a, doc
OPEN AND CLOSED SETS
Some "Geometric" Definitions
I) The set consisting ofall xe R" such that Ix-xol 0 such that Dra )CA
Since n is open, there cxists ra o such illa D,()GD. Now, by taking r min (ru.ra)
Theorem
Let A CR the setA contains at least one of its boundary points, then A is not open
Definition
A set AgR" is said to be closed if it contains all of its boundary points.
R are both open and closed.
LIMITs IN R
Definition: The General Case
let A c R" R where A is open. Let ac A or a boundary point of A.Then Iim f(N)-L means for
each open set Ng R" that contains L there exists an open set M GR" that contains a such that
MAT23TY1a doc
Example
Let f
The Algebra of Limits
3 lf lim f(x) L and lim f (x) M, then L-M
4) Let c R" R" g:BeR" RP, where A. B. f(A)nB are open. If linn. (x) Le Band
lim gly) M, then lim(g of XN)- M
Exercise
Evaluate the limit if it exists
Approaching (0.0) along (r.r)., li
along (r.2), lim slim So the limit doesn'texists.
Using polar coordinates.
0 Alternately, we know
CONTINUITY
Defintion
Let f
g R" R" fis said to be continuous at a e A if
I) (a) is defined
Page of 23
MAT237 Y1a, doc OPEN AND CLOSED SETS Some "Geometric" Definitions I) The set consisting ofall xe R" such that Ix-xol 0 such that Dra )CA Since n is open, there cxists ra o such illa D,()GD. Now, by taking r min (ru.ra) Theorem Let A CR the setA contains at least one of its boundary points, then A is not open Definition A set AgR" is said to be closed if it contains all of its boundary points. R are both open and closed. LIMITs IN R Definition: The General Case let A c R" R where A is open. Let ac A or a boundary point of A.Then Iim f(N)-L means for each open set Ng R" that contains L there exists an open set M GR" that contains a such that MAT23TY1a doc Example Let f The Algebra of Limits 3 lf lim f(x) L and lim f (x) M, then L-M 4) Let c R" R" g:BeR" RP, where A. B. f(A)nB are open. If linn. (x) Le Band lim gly) M, then lim(g of XN)- M Exercise Evaluate the limit if it exists Approaching (0.0) along (r.r)., li along (r.2), lim slim So the limit doesn'texists. Using polar coordinates. 0 Alternately, we know CONTINUITY Defintion Let f g R" R" fis said to be continuous at a e A if I) (a) is defined Page of 23LAT237Y1a doc
Theorem
If P(x) and o(N) are polynomials, then lim rA-Hil.o(a)20
Example
is continuous everywhere.
Differentiation
DERIVATIVE
Definition: Derivative of a Single-Variable Function
Let f
limit. if itexists, is usually denoted f(a) and is called the derivative of s() at a
If is diffaentiable for all xe I. then the function str) Em
is usually called the
derivative of the function f
Definition: The Partial Derivative of a Two-Variable Function
pen, (a
exists. then it is called the partial
derivative of the function f with respect lovat .t) andis denoted h)
Geometrically
(a, b) is just the slope of the curve of intersection of the surface z- jlr.) with respect!othe plane
y b at the point (a, b, f(a,b)
In general. if z y) isasurface for which a "rangent plane exists at the point (a.b.f(a,b).then the
equation for the tangent plane is (a, bXx-a)+ bXy-a)-(z-fu,b) 0
Application
lf z f(r y) has a tangent plane at the point (a,b, f(a, b) and and A are "smalr numbers, then
Page of 23
MAT237Y1a doc
Definition: Partial Derivative in General
it is the partial derivative off with respect to r
Some Properties of the Derivative
I) If f:R" R g:R" R", aand pare constants, then Dlad t Ak) aDf
3) If f:R" R g:R" R, then v
DIFFERENTIABILITY
Definition: Differentiability of Two Variablo Functions
Let f:A s R2 R.We say f is differentiable at (a,b)EA if
af
Definition: Differentiability in General
Let f:AGR" R.We say fis differentiable at a e A if all the partial derivatives associated with fata
0, where Tis the matrix of partial derivatives associated with fat a.
PATHS IN R
Definition
If denotes an interval in R, then the function f:ISR R" is called a path in R
Example
s:lo4r] R 3. H (cost sin) is a circular path traveled twice.
Definition
(a) (r(al v Talz (a) is called the velocity vector the path at r a
Page 10 of 23
LAT237Y1a doc Theorem If P(x) and o(N) are polynomials, then lim rA-Hil.o(a)20 Example is continuous everywhere. Differentiation DERIVATIVE Definition: Derivative of a Single-Variable Function Let f limit. if itexists, is usually denoted f(a) and is called the derivative of s() at a If is diffaentiable for all xe I. then the function str) Em is usually called the derivative of the function f Definition: The Partial Derivative of a Two-Variable Function pen, (a exists. then it is called the partial derivative of the function f with respect lovat .t) andis denoted h) Geometrically (a, b) is just the slope of the curve of intersection of the surface z- jlr.) with respect!othe plane y b at the point (a, b, f(a,b) In general. if z y) isasurface for which a "rangent plane exists at the point (a.b.f(a,b).then the equation for the tangent plane is (a, bXx-a)+ bXy-a)-(z-fu,b) 0 Application lf z f(r y) has a tangent plane at the point (a,b, f(a, b) and and A are "smalr numbers, then Page of 23 MAT237Y1a doc Definition: Partial Derivative in General it is the partial derivative off with respect to r Some Properties of the Derivative I) If f:R" R g:R" R", aand pare constants, then Dlad t Ak) aDf 3) If f:R" R g:R" R, then v DIFFERENTIABILITY Definition: Differentiability of Two Variablo Functions Let f:A s R2 R.We say f is differentiable at (a,b)EA if af Definition: Differentiability in General Let f:AGR" R.We say fis differentiable at a e A if all the partial derivatives associated with fata 0, where Tis the matrix of partial derivatives associated with fat a. PATHS IN R Definition If denotes an interval in R, then the function f:ISR R" is called a path in R Example s:lo4r] R 3. H (cost sin) is a circular path traveled twice. Definition (a) (r(al v Talz (a) is called the velocity vector the path at r a Page 10 of 23MAT237 Y1a.doc
The Algebra of Path Velocities
Let f.
be paths. aand pconstants, h: R Ran ordinary function. Then
(g of
f(a))If
called the gradient.
THE CHAIN RULE IN GENERAL
Let f A CR
R". g:B cR" RP.ac A such that fa)e B.If is differentiable at a and is
differentiable at y fa), then ID (g of
a) [DRXf(a)IDyla)]
Useful Property
If F(ri.....x.) o implicitly defines each of the variables asa function of the remaining variables
DIRECTIONAL DERIVATIVE
Definition
Let S:AgR3 R a (a b)e A. and u (ul.u2) unit vector. If lim
exists, then it is
called the directional vector of the function fat the point (a,b) in the direction of the vector u. It is denoted
Theorem
Dus(a) Vf (a), u where v
the gradient.
Example
Page 11 of 23
MAT237Y1a doc
Consider a domed roof with the shape of the surface z 15- -2y A marble is placed at (3,1,4). In what
direction will the marble fall!
f(3,
V52 cos(vj
So the marble will fall in the direction 3.2)
Theorem
Theorem
v (N) points in the direction of maximum increase of the functionfar the point x.
2 -V (x) points in the direction of maximum decrease of the functionfat the
point x
Theorem
lfs is the level surface given by the equation Flr,y.z) k and s has a tangent plane at (ro. ye.zo). then
wFlro. yo.zo) is a normal vector to the tangent plane lo sat the point (ro.yo.zo).
Proof: We just show that if ri)denotes any path on the surface sthat passes through (ro.yo.zo), the our
gradient vF(ro. yo.to) is perpendicular to the tangent vector of ri) at (wo, y..zo). So let rh) be acurve
F dF
Higher-Order Derivatives
THE T
OR THEOREM
First order Taylor Expansion
Theorem
Page 12 of 23
MAT237 Y1a.doc The Algebra of Path Velocit

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