MATH 14 Study Guide - Fall 2019, Comprehensive Midterm Notes - Theorem, Integral, Dimension
14 Feb 2019
School
Department
Course
Professor
MATH 14
1/7/19
Lecture 1: Intro
Table of Contents
Ⅰ) Math 13 & Math 14 Compare and Contrast 1
Ⅱ) n dimensional projects in m dimensional space (where nm). 1
Ⅲ) Physical Questions 3
Ⅰ) Math 13 & Math 14 Compare and Contrast
●Math 13 constant vectors: v = 3i – 7j + 14k
vs. Math 14 vector functions: v is a function of time.
e. g. v(t
) = t
2i + et
j + ln(t
4)k
This is very closely related to parameterized paths from Math 13.
●Math 14: vector fields: v is a function of spatial variables.
v(x, y, z) = x2yi + j + cos(xeyz)kexz2
e. g. v = ▽f(x, y, z) from Math 13.
Notation: F, not v, for vector fields.
Ⅱ) n dimensional projects in m dimensional space (where
).n ≤m
It’s crucial to able to classify them, especially at the end of the Math 14 course.
m =
1–D Space
2–D Space
3–D Space
1 out of 3
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1–D Object
e.g. straight wire
e.g. location of a
car on a straight
road: x = f
(t
).
e.g. curved wire
e.g. parameterized
path of an ant on a
table.
x = f
(t
)
y = g
(t
)
e.g. circumference
of a circle:
x2 + y2 = 1
e.g. curved wire
e.g. parameterized
path of a baby sent
down a really cool
water slide.
x = f
(t
)
y = g
(t
)
z = h
(t
)
2–D Object
none
e.g. disk filled in
circle like x2 + y2
. ≤ 1
e.g. paraboloid
z = f
(x, y) = x2 +
y2
e.g. surfaces of a
sphere like x2 + y2
+ z2 = 1.
3–D Object
none
none
e.g. brick or a
filled in box.
e.g. a ball (filled in
2 out of 3
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spheres).
x2 + y2 + z2 . ≤ 1
Ⅲ) Physical Questions
A. 1-D Objects (next 6 lectures).
1. Length of a wire (even if curved)
2. Mass of a wire (even if curved, even if density varies along the wire).
3. Center of mass for a wire (even if curved, even if density varies along
the wire).
4. Work done on a moving object by a vector field representing force.
B. 2-D & 3-D objects (ch. 15, next 8 lectures)
1. Mass of an object (even if density varies in the object).
2. Center of mass of an object (even if density varies in the object).
C. Flux of a vector field through…(2 lectures)
1. A 2-D surface.
❖e.g. How much solar energy goes through a window? Through a
dome?
2. ...a 1-D curve
D. Extending the notion of the gradient of a function to the gradient of a field
1. Divergence of a field
2. Curl of a field
...and extending part 2 of the fundamental theorem of calculus
to 2-D & 3-D:(x)dx (b) (a)
∫
b
a
f′=f−f
1. Divergence theorem
2. Stoke’s theorem (Green’s thm.)
Gradient: ▽f Div: ▽f· Curl: ▽f×
3 out of 3
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Document Summary
) math 13 & math 14 compare and contrast. ) n dimensional projects in m dimensional space (where nm). Math 13 constant vectors: v = 3 i 7 j + 14 k vs. Math 14 vector functions: v is a function of time: g. v ( t ) = t 2 i + e t j + ln( t 4 ) k. This is very closely related to parameterized paths from math 13. Math 14: vector fields: v is a function of spatial variables. exz2 v (x, y, z) = x 2 y i + j + cos(xe y z) k: g. v = f(x, y, z) from math 13. ) n dimensional projects in m dimensional space (where n m. It"s crucial to able to classify them, especially at the end of the math 14 course. m = 1 d object e. g. straight wire e. g. location of a car on a straight road: x = f ( t ).
