TRIGONOMETRY
Right Triangle Definitions opp adj sin = cos = hyp hyp opp adj cot = tan = adj opp hyp hyp sec = csc = adj opp Circular Definitions
sin = tan = sec = y r y x r x cos = cot = csc = x r x y r y tan x = sec x = sin x cos x 1 cos x cot x = csc x
PHYS1315 Methods in Physics I
2012/13 Homework 5 Solutions
1. (a) Rewrite the given DE in the form of
where
We find that
Let
,
.
, this DE is an exact equation.
be the function such that
So we have:
Comparing the above equations, we find:
Therefore the ge
PHYS1315 Methods in Physics I
2012/13 Homework 5
(Due date: 7 Dec 2012, 2:30 pm)
1. Identify each of the following equations as separable, linear, homogeneous, or exact.
Then use appropriate method to nd the general solution.
(a) (y + 2xy 3 ) dx + (1 + 3x
PHYS1315 Methods in Physics I
2012/13 Homework 4 Solutions
1. (a) To find the intersecting line L of these two surface, we draw the cross-section of
the solid on the plane y = 0 as shown below.
If y = 0, the equation of the surfaces become:
We can see tha
PHYS1315 Methods in Physics I
2012/13 Homework 4
(Due date: 23 Nov 2012, 2:30 pm)
1. Use double integrals in polar coordinates to nd the volume of the solid region described
below.
(a) The solid that is bounded by the sphere x2 + y2 + z2 = 4 and the parab
PHYS1315 Methods in Physics I
2012/13 Homework 3 Solutions (amended)
1. We first calculate the velocity vector for the curve:
and the gradient of f(x, y, z):
The point on the curve when
is:
At this point,
It implies that
is the scalar multiple of
mentione
PHYS1315 Methods in Physics I
2012/13 Homework 3
(Due date: 7 Nov 2012, 2:30 pm)
1. A smooth curve is normal to a surface f (x, y, z ) = c at a point of intersection if its
velocity vector is a nonzero scalar multiple of f at that point.
Show that the cur
PHYS1315 Methods in Physics I
2012/13 Homework 2 Solutions
1. (a) Speed
Acceleration
(b) Position at time t:
(c) Unit tangent vector:
Recall that the acceleration
Tangential component of acceleration:
Normal component of acceleration:
(d) Curvature of the
PHYS1315 Methods in Physics I
2012/13 Homework 2
(Due date: 19 Oct 2012, 2:00 pm)
1. Suppose a particle moving along a curve C in space has the velocity
v(t) = cos t sin t + et k .
i
j
(a) Find the speed and acceleration of the particle at time t.
(b) If
PHYS1315 Methods in Physics I
2012/13 Homework 1 Solutions
1. (a) The volume can be calculated by
So it goes to:
.
Hence,
(b) The area of the face can be calculated by
So it goes to
(c) To find out the angle between u and the plane containing the face det
PHYS1315 Methods in Physics I
2012/13 Homework 1
(Due date: 5 Oct 2012, 2:00 pm)
1. Consider the parallelepiped with adjacent edges:
u = 4 8 + k, v = 2 + 2k, w = 3 4 + 12k
i
j
ij
i
j
(a) Find the volume of the parallelepiped.
(b) Find the area of the face
PHYS1315 Methods in Physics I
2012/13 Solutions to Exercise 7
1. (a)
x
W (e , ln x; x) =
ex ln x
ex 1/x
=
1
ln x
x
= 0 x [1, )
It implies that ex and ln x are linearly independent in the interval x (1, ).
(b)
W (sin x, tan x; x) =
sin x
tan x
2
= sin x(s
PHYS1315 Methods in Physics I
2012/13 Exercise 7
1. Use the Wronskian to determine whether the given functions are linearly independent
on the given interval.
(a) ex , ln x; [1, )
(b) sin x, tan x; [0, /3]
(c) x2 , 2x 5x2 , x; (, )
2. (a) Show that y1 (x)
PHYS1315 Methods in Physics I
2012/13 Solutions to Exercise 6
1. (a) The integrating factor of the given dierential equation is:
2
1
(x) = exp
dx = 2
x
x
Thus the equation can be reduced to:
1 dy 2y
x+1
3=
2 dx
2
x
x
x
x+1
dy
=
dx x2
x2
x+1
y
dx + C
2=
PHYS1315 Methods in Physics I
2012/13 Exercise 6
1. Solve the following dierential equations by using the integration factors.
dy 2y
= x+1
(a)
dx
x
dy
2x + 1
(b)
+
y = e2x
dx
x
dx
(c)
+ x cot t = 2t csc t
dt
2. Solve the following dierential equations by
PHYS1315 Methods in Physics I
2012/13 Solutions to Exercise 5
1. (a) Let F = (x + 2y ) + (x y ) = P + Q and the path C : r(t) = 2 cos t + 4 sin t ,
i
j
i
j
i
j
0 t /4.
/4
dx
dy
P
F dl =
+Q
dt
dt
dt
0
C
/4
[(2 cos t + 8 sin t)(2 sin t) + (2 cos t 4 sin t
PHYS1315 Methods in Physics I
2012/13 Exercise 5
1. Evaluate the line integral for each of the following vector elds along the given path.
(a) F( x, y) = ( x + 2y) + ( x y) along the curve r(t) = 2 cos t + 4 sin t for 0 t /4.
i
j
i
j
(b) F( x, y) = (y x)
PHYS1315 Methods in Physics I
2012/13 Solutions to Exercise 4
3
1
1. (a)
1
0
2xy
dx dy =
x2 + 1
/3
3
3
2
y [ln(x +
1)]1
0
1
/3
/2
(b)
dy =
1
(x sin y y sin x) dx dy =
0
0
0
/3
=
0
1
y ln 2 dy = y 2 ln 2
2
1
y cos x + x2 sin y
2
1
y + 2 sin y
8
1
1
= y
PHYS1315 Methods in Physics I
2012/13 Exercise 4
1. Evaluate the double integrals given below.
1
3
2xy
(a)
dx dy
2
0 x +1
1
/3
/2
(x sin y y sin x) dx dy
(b)
0
1
0
2
4 cos x2 dx dy
(c)
0
8
(d)
2y
2
3
0
y4
x
1
dy dx
+1
2. Compute the following double int
PHYS1315 Methods in Physics I
2012/13 Exercise 3
1. Determine whether the limit exists. If so, nd its value.
x4 y 4
(a)
lim
(x,y )(0,0) x2 + y 2
xy
(b)
lim
2 + 2y 2
(x,y )(0,0) 3x
xz 2
(c)
lim
(x,y,z )(2,1,2)
x2 + y 2 + z 2
2. Let
f (x, y ) =
0, xy = 0
1,
PHYS1315 Methods in Physics I
2012/13 Solutions to Exercise 2
1. (a) Let y = t. Then
x = t2 , z = 9 t2 t4 .
Thus the curve of intersection is represented by:
i
j
F(t) = t2 + t + 9 t2 t4 k
where ( 37 1)/2 t ( 37 1)/2. Note that the range of possible
values
PHYS1315 Methods in Physics I
2012/13 Exercise 2
1. Find the vector function F(t) representing each of the following curves.
(a) The curve of intersection of the hemisphere z = 9 x2 y 2 and the parabolic
cylinder x = y 2 .
(b) The line of intersection of
PHYS1315 Methods in Physics I
2012/13 Solutions to Exercise 1
1. (a) y = sin x
(b) 2x + y + 3z = 6
(c) z = ey
2. (a)
(b)
(c)
(d)
(y 2)2 + z 2 = 4 and x = 0.
y = 3 and z = 1.
0 x 2, 0 y 2, and 0 z 2.
1 x2 + y 2 + z 2 4.
3. Given:
AB + BC + CD + DA = 0,
PHYS1315 Methods in Physics I
2012/13 Exercise 1
1. In each part, sketch the surfaces in 3D space.
(a) y = sin x
(b) 2x + y + 3z = 6
(c) z = ey
2. Write equations or inequalities to describe each of the given sets.
(a) The circle of radius 2 centered at (
Appendix B Taylor Series and Extreme Values of Multivariable Functions
Page 1
Appendix B Taylor Series and Extreme Values
of Multivariable Functions
B.1 Taylor Series in Several Variables
Functions of two or more variables can be often expanded as a power
Appendix A Triple Products
Page 1
Appendix A Triple Products
Definition: Scalar Triple Product
The product A (B C) is known as the scalar triple product of A, B, and C. The scalar
triple product of A, B, and C is usually written simply as [A, B, C], and i
PHYS1315 Methods in Physics I
Quick Review of Calculus
I
Limits and Continuity
I.1
Evaluating Limits
Denition: Let f (x) be dened on an open interval about x0 , except possibly at x0 itself.
If f (x) gets arbitrarily close to L for all x suciently close t