Example 1. We need an example to illustrate
whats going on. Let f (x, y ) =
x2 + y 2 , let
x(s, t) = s ln t, and let y (s, t) = sin x + cos t. Then
f (x(s, t), y (s, t) =
The chain rule, part 2
Math 131 Multivariate Calculus
(s ln t)2 + (sin s + cos t)2 .
Now lets add limits of integration. If the limits
of integration for u are a and b, then the limits of
integration for x will be x(a) and x(b).
x(b)
Jacobians
Math 131 Multivariate Calculus
b
f (x) dx =
x(a)
f (x(u)
a
dx
du.
du
D Joyce, Spring 2014
We wan
and its derivative is the acceleration,
a(t) = v = x = (x , y , z ).
Now, Kepler determined that the acceleration of
the planet was toward the sun. Newton explained
that in terms of gravitational force by saying, rst,
that the gravitation of the sun is an
the standard (x, y, z ) coordinate three-space. Notations for vectors. Geometric interpretation as vectors as displacements as well as vectors as points.
Vector addition, the zero vector 0, vector subtracThings you need to know about
tion, scalar multipli
About some topological concepts. In the
text theres a discussion of various topological
conceptsopen, closed, boundary, neighborhood,
and accumulation point. These are particularly important when the concepts of limit and continuity
are extended to more g
Theorem 1. When the rst and second partials of
a function of two variables x and y are all continuous functions, the order that the partials are taken
doesnt matter, that is, fyx = fxy , or in partial notation
2f
2f
=
.
x y
y x
Higher-order partial deri
Surface integrals
Math 131 Multivariate Calculus
D Joyce, Spring 2014
The area dierential of a surface, and a double integral for the area of the surface.
Recall that were using X(s, t) to describe a paramaterization of a surface S in 3-space. Also
we hav
Whereas a line is parametrized by one variable
weve usually used ta surface is parametrized by
two variableswell usually use s and t.
It helps to have a standard notation. Well use S
to denote the surface in R3 that were parametrizSurfaces
ing. Well use X
Note how y and z dont change in this limit.
This kind of derivative is called a partial derivative since only one of the variables changes. Furthermore, the concept of limit used here is just the
Partial derivatives
scalar limit you used in calculus, not
A Lorenz attractor. Math 131 Multivariate Calculus
D Joyce, Spring 2014
A Lorenz attractor. Sometimes surprising things can happen. Lets start with this vector
eld in space.
8
F(x, y, z) = (10(y x), 28x y xz, 3 z + xy)
It seems pretty innocuous since its
is a good approximation of f near a in the sense
that
f (x) h(x)
= 0.
lim
xa
xa
Note that when m = 1, this denition says the
derivative Df of the scalar-valued function f is the
1 n row-matrix
Total derivatives
Math 131 Multivariate Calculus
D Joyce, Spri
speed, x , to be the length of velocity, so speed is
a nonnegative scalar.
So, at a time t, an object has a position x = x(t);
a velocity x = x (t), which is sometimes denoted
v = v(t); an acceleration x = x (t), which is
sometimes denoted a = a(t); and a
system mentioned above has these three equations
fx (x, y ) = 2y = gx (x, y ) = 2x
fy (x, y ) = 2x + 2y = gy (x, y ) = 2y
g (x) = x2 + y 2 = c = 1
Lagrange multipliers
Math 131 Multivariate Calculus
in the three unknowns x, y , and . The rst two
equations
Gradients
Math 131 Multivariate Calculus
D Joyce, Spring 2014
f
Last time. Introduced partial derivatives like
x
of scalar-valued functions Rn R, also called
scalar elds on Rn .
Total derivatives. Weve seen what partial
derivatives of scalar-valued functi
Onto functions (surjective), one-to-one functions (injective), one-to-one correspondences
(bijective), range. These terms mean the same
thing in multivariate calculus as they in calculus of
a single variable. Be sure youre familiar with the
following conc
Proof. One direction is easy. Suppose that F has
path-independent line integrals. Let C be a simple
closed curve, starting and ending at the same point
a. Another path with the same endpoint is the
constant path, which we can denote a. Since C
and a have
and well usually take the variable to be t suggesting
time. The curve is the image of this path, that is,
a subset of Rm .
Examples 1 (Unit circle). For instance, v : R
R2
x(t) = (cos t, sin t)
Curves and paths
Math 131 Multivariate Calculus
D Joyce, Spr
dx
since we saw earlier that
was equal to x .
x
ds
The unit tangent vector T gives the direction of
the curve. Its useful because it says which way
the path is going, but doesnt indicate how fast
The unit tangent vector
the object is travelling that path.
always works to give the total distance L travelled
along the x-axis even when the object alternates
the direction it moves along the x-axis.
By analogy, you would guess that even when the
direction of travel is not restricted to the x-axis,
the distance
Example 1 (A maximum). Let the potential function be
f (x) = x2 y 2
which is illustrated in gure 1.
It clearly has a maximum at (0, 0). The equipotential curves are concentric circles x2 + y 2 = C
with center at (0, 0). The gradient eld is F(x, y) =
f (x,
Therefore,
=
Directional derivatives, steepest
ascent, tangent planes
Math 131 Multivariate Calculus
=
=
D Joyce, Spring 2014
=
Directional derivatives. Consider a scalar eld
f : Rn R on Rn . So far we have only considered
the partial derivatives in the d
Unit vectors. A unit vector is a vector whose
length is 1. We can interpret unit vectors as being
directions, and we can use them in place of angles
since they carry the same information as an angle.
Unit vectors can be identied with points on the
unit sp
If the second derivative f (a) was 0, then you had
to try some other way to determine whether it
was a max or min or neither. We need some sort
of second-derivative test for the multidimensional
case.
Maxima and minima of scalar elds
Math 131 Multivariate
The surface x2 + y 2 + 1 = z is a paraboloid opening
upward (positive z being upward) with vertex on
the z -axis at z = 1. Above that surface and below
the plane z = 5 lies the 3-dimenional region D. The
top surface of D is a circle of radius 2. Let F be
Here, the symbol is just a variant of thats often
used for line integrals when the line is a closed curve
or a nite union of closed curves.
Greens theorem
Math 131 Multivariate Calculus
Example 1. Let F = (M, N ) = (2y, x) and D is
the semicircular region
D
M
dA as follows.
y
M
dA
y
D
b
(x)
M
dy dx
y
a
(x)
Proof of Greens theorem
Math 131 Multivariate Calculus
=
D Joyce, Spring 2014
(x)
b
M (x, y)
=
a
Summary of the discussion so far.
b
M (x, (x) M (x, (x) dx
=
M
N
x
y
M dx + N dy =
D
D
dx
y=(x)
a
b
dA.
b
Example 1 (A radial vector eld). Well look as
several vector elds in the plane, and draw them
by drawing a few vectors F(x) with their tails attached to the points x. For instance, the vector
eld F(x) = (1, 1) is a constant vector eld with all
Intro to ve