LINE AND 17 SURFACE INTEGRALS
17.1 Vector Fields (ET Section 16.1)
1. Which of the following is a unit vector eld in the plane? (a) F = y, x (b) F = (c) F = y x 2 + y2 , x x 2 + y2
y x , x 2 + y2 x 2 + y2
(a) The len
DIFFERENTIATION IN 15 S E VERAL VA RIABLES
15.1 Functions of Two or More Variables
1. What is the difference between a horizontal trace and a level curve? How are they related?
SOLUTION A horizontal trace at height c consists of
Vector equations and Vector equations of multiple variables
Lets write the equation of a line as a vector equation in terms of r (t )
The equation is
3x 1 y z 1
What is the domain of the function?
Lets look at the domain of the following v
Vector/Parametric Equations and Lines
Given the following linear equations, write parametric equations, vector equations and
symmetric equations for each.
1) 3x + 5y = 15
2) 4y = 16
Vector Equations _
9) Given the planes 2 y + z = 0 and
3x + y + z = 2
a) Find the angle between the planes
b) Find the equation of the line of intersection of the planes
c) What is the distance from the point A(3,2,-4) and the second plane?
Notes over Hyperbolic Trig functions
e x e x
e x + e x
e x e x
e x + e x
Using these relationships, find the following:
Find which hyperbolic trig functions are even and odd and show their relationship.
1) sinh( ) =
Linear Differential Equations
Solving a Linear differential equation:
Given the differential equation given g(t) and p(t) are continuous functions.
Lets assume that there is a function that exists named and we will call it the integrating factor.
We can m
Separable Differential Equations
We worked separable differential equations back in BC Calculus. The only item to add will be
the requirement that the solution to our differential equation must be a continuous equation.
Notes over Unit Vector, Tangent Vector, Unit Tangent Vector.
Method for finding Unit Vector: Unit Ve c t o r f o r v =
Method for finding Tangent vector: G ive n v =< f ( x ) , g ( x ) , h ( x ) >
T h e t ang e nt ve c t o r v ' =< f ' (
Multivariable Calculus Triple Integral WS
1) Evaluate the following triple integrals as iterated integrals.
x y d V
wh e r e E = [ 0 , 1] x [ 0 , 2 ] x [ 0 , 3]
z 2 d V wh e r e E = [ 1, 1] x [ 3, 3] x [ 0 , 3]
z c o
Multivariable Test Review Gradients
1) f ( x, y ) = sin( xy ) + cos( xy )
2) f ( x, y , z ) = 2 x y + 2 y z +
x = r cos
x = et
y = r sin
y = es z = s + t
3) At what points does x2 y2 + 4xy = 4 have vertica
x = sin cos
y = sin sin
z = cos
2 = x2 + y 2 + z 2
The symbols represent the following:
= distance from the origin to the point.
= angle drawn from the z axis down to the point.
= angle drawn rrom the positive x a
Spherical and Cylindrical WS Name _
1) What is the following implicit equation the graph of?
z = x2 + y 2
2) Sketch the graph in the domain of [-10,10] and [-10,10] on the xyz system.
3) Write the equation of problem #1 in cylindrical coordinat
Sage Velocity and Acceleration WS Name _
1) Given a position vector s(t ) = e , t 1, t + 4 find the following.
a) Velocity vector
b) Acceleration vector
c) Representation of each of the 3 vectors when t = 2 on a 3D coordinate axis in
Sage partial Derivative WS
1) f ( x, y ) = x 2 y + y 2 Find the matrix of partial derivatives for f(x,y).
2) f ( x, y ) = x 2 y + y 2 Find the matrix of second partial derivatives for f(x,y).
3) f ( x, y ) = e x cos y Find the matrix of partial derivative
1) Given the function f ( x , y ) =
over the domain of x [-2,2] and y [0,3]
+ 1) ( y 2 + 1)
a) Find the equation of the equation of the normal line(in parametric form) at (1,1,1)
b) Find the equation of the tangent plane at the point (1,1
Sage Implicit Diff. WS
Process for finding derivative using implicit differentiation.
Define your y-variable using the var command
Define a function as an implicit function of x using the function ( ) command (yf)
Move everything to
Answer each of the questions on your own paper, and be sure to show your work
so that partial credit can be adequately assessed. Put your name on each page of
You may use a scientic calculator, but it should not
Math 2057-5 Quiz #2 (Fall 2005)
1) Let z = x2 + xy + y 2 , x = s + t and y = st.
a) Use the chain rule to nd z/s = (2x + y ) + (x + 2y )t = (2 + t)x + (1 + 2t)y
Solution: Note that z is a function of the independent variables s and t:
Math 2057-5 Quiz #1 (Fall 2005)
1) Let f (x, y ) = x ln(3x + 2y ), 3x + 2y > 0. Find the partial derivative fxy .
(3x + 2y )2
First we have to nd the partial derivative fx , and then we have to dierentiate fx with
respect to the variab
Math 2057-5, Test # 2. Fall 2005
D = fxx (x, y )fyy (x, y ) (fxy (x, y )2 .
1) Let f (x, y ) = ln(x2 + y 2 ).
a[10P]) Find the directional derivative of f at the point P (2, 1) in the direction of the vector
v =< 3, 4 >.