FUNCTIONS OF SEVERAL VARIABLES: GRADIENTS [SST 11.6]
GRADIENTS:
Let f (x, y) C (1,1) .
Then the gradient of f is a vector in R2 given by:
grad f
Let f (x, y, z) C (1,1,1) .
f (x, y)
f :=
f f
,
x
Matthew Dehlinger
Assignment 11.1 due 12/09/2014 at 11:59pm CST
1. (6 pts) Suppose
values:
f (4, 5)
f (5, 4)
f (0, 0)
f (5, 5)
f (t, 4t)
f (uv, u v)
f14jengwerm2450s020
f (x, y) = xy2 5. Compute the f
Matthew Dehlinger
Assignment 9.7 due 12/09/2014 at 11:59pm CST
f14jengwerm2450s020
4. Empty set
1. (1 pt) State whether the equation
4x2 + 49y2 9z2 = 1
denes (enter number of statement):
1. A hyperbol
Matthew Dehlinger
Assignment 13.2 due 12/09/2014 at 11:59pm CST
f14jengwerm2450s020
1. (6 pts) Determine whether the line integral of each vector
eld (in blue) along the semicircular, oriented path (i
GREENS THEOREM [SST 13.4]
CLOSED CURVES:
A closed curve is a curve that begins & ends at the same point.
Special notation is used with line integrals along closed curves:
f ds
f dx
C
F dR
f dy
C
C
C
EX 13.2.6:
Find the work done by the force eld F(x, y, z) = 2xy, x2 + 2, y on an object moving along ,
where is the line segment from (1, 0, 2) to (3, 4, 1).
Recall the line integral formula for work
EX 13.3.1:
(a)
Let vector eld F(x, y) = y 2 , 2xy 3 and C be any smooth path from P (0, 0) to Q(2, 1).
Verify that F is conservative.
(b)
Find a scalar potential f for F.
(c)
F dR.
Compute I =
C
(a) L
1 + x2 + y 2 + z 2 harmonic in R3 ? (Justify answer)
EX 13.1.6: Is f (x, y, z) =
Recall that a function is harmonic if its Laplacian equals zero.
1st , compute the rst-order partials of f :
1
1
1/2
f
EX 13.2.3:
xy 2 dx + x2 dy , where C is the path from (1, 1) to (0, 1) along line y = 1 and then from
Compute
C
(0, 1) to (1, 0) along quarter-circle x2 + y 2 = 1.
Path C is too hard to parametrize!
MATH 2450-020: EXAM 3 INFO/LOGISTICS/ADVICE
INFO:
WHEN:
Wednesday (11/05) at 1:00pm in PETRE 121 (our usual room)
DURATION:
50 mins
PROBLEM COUNT:
Appropriate for a 50-min exam
BONUS COUNT:
Several
EX 13.6.2:
Let curve C be the intersection of plane y + z = 2 & cylinder x2 + y 2 = 1 oriented CCW as viewed from above.
y 2 dx + x dy + z 2 dz .
Use Stokes Theorem to compute the line integral I =
C
EX 13.3.3:
(a)
t
2
Let C be the path traced by R(t) = 2 sin
Verify that line integral I =
cos(t), arcsin t for 0 t 1.
(sin y) dx + (3 + x cos y) dy is independent of path (IoP).
C
(b)
Compute line int
EX 13.7.2:
Let solid E be bounded by the planes z = 0, y = 0, y = 2, and the parabolic cylinder z = 1 x2 .
Let surface S be the boundary of solid E with outward unit normal N.
Let vector eld F(x, y, z
Matthew Dehlinger
Assignment 13.2 due 12/09/2014 at 11:59pm CST
f14jengwerm2450s020
1. (6 pts) Determine whether the line integral of each vector
eld (in blue) along the semicircular, oriented path (i
Matthew Dehlinger
Assignment 13.3 due 12/09/2014 at 11:59pm CST
f14jengwerm2450s020
1. (12 pts)
f=
Show that the vector eld F(x, y, z) = y sin(z) i + (x sin(z) +
2y) j + (xy cos(z) k is conservative b
LINE INTEGRALS [SST 13.2]
LINE INTEGRAL OF A SCALAR FIELD:
x = x(t)
Given smooth curve C :
y = y(t) and continuous scalar eld f (x, y)
t [a, b]
Then the line integral of f over curve C is dened to
VECTOR FIELDS: INTRO, DIV, CURL [SST 13.1]
THE FUNCTION LANDSCAPE:
FUNCTION TYPE
PROTOTYPE
MAPPING
(Scalar) Function
y = f (x)
f :RR
2D Vector Function
F(t) = f1 (t), f2 (t)
F : R R2
3D Vector Functi
SURFACE INTEGRALS & FLUX INTEGRALS [SST 13.5]
SMOOTH SURFACE:
A surface S is smooth if the normal vector at each point in S exists & is nonzero.
PIECEWISE SMOOTH SURFACE:
A surface S is piecewise sm
GRADIENT FIELDS, SCALAR POTENTIALS, PATH INDEPENDENCE [SST 13.3]
GRADIENT FIELD (CONSERVATIVE VECTOR FIELD):
F is a gradient eld F is conservative F =
f for some scalar eld f f is a scalar potential
Matthew Dehlinger
Assignment 12.7 due 11/06/2014 at 11:59pm CST
f14jengwerm2450s020
1. (10 pts)
Match the integrals with the type of coordinates which make
them the easiest to do. Put the letter of th
EX 12.7.5:
z dV , where E is the solid bounded above by plane z = 3 &
Using spherical coordinates, compute I =
E
x2 + y 2 .
below by the half-cone z =
1st , although not required (and will not be grad
3
EX 12.2.8:
4y
Let I =
f (x, y) dx dy.
0
y/3
Sketch the region of integration & write an equivalent iterated double integral with the order of integration reversed.
Since the given order is dx dy, t
EX 12.5.6:
Using a triple integral, nd the volume of the solid E bounded above by the surface x2 + y 2 + z 3 = 9 and below
by the plane z = 0.
1st : Intersect surface (x2 + y 2 + z 3 = 9) with plane (
Use the inverse transformation T 1 :
EX 12.8.4:
u = xy
v = x2 y 2
to setup
dA,
D
where D is the region in Quadrant I bounded by the hyperbolas xy = 1, xy = 3, x2 y 2 = 1 & x2 y 2 = 4.
1st , sketch reg
DOUBLE INTEGRALS: RECTANGULAR COORDINATES [SST 12.2]
PROPERTIES OF DOUBLE INTEGRALS:
Let set D be a closed & bounded region in R2 .
Let functions f (x, y) & g(x, y) be dened on D.
Let k R.
Constant
DOUBLE INTEGRALS: POLAR COORDINATES [SST 12.3]
SPECIAL POLAR CURVES:
a = 0, b = 0, k = 0, n Z+
Z+ The set of all positive integers.
POLAR CURVE
PROTOTYPE
REMARK(S)
Lines (Rays) thru Pole
=k
Always Gr