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 1
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
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 forc
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 sub
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
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
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
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 sur
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
Part
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), 2
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 d
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 acceleratio
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
i
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. Wev
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 v
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
co
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)
Curve
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
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 o
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
wit
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 w
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.
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.
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.
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
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
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)