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Section(s)
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Concepts to Master
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13.1
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Rectangular three-dimensional coordinates
Distance between two points
Planes, spheres, and regions in space
This
section
extends the two dimensional x-, y- coordinate system to three
dimensions (x,
y, z) with three axes, each perpendicular to the others as you
might visualize
the corner of a room.
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13.2
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Vectors; Position vector; Length: Unit
vector: Standard basis vector (i, j, k)
Vector arithmetic; scalars; Vector properties
This
section extends the two dimensional x-, y- coordinate system to three
dimensions (x, y, z) with three axes, each perpendicular to the others as you
might visualize the corner of a room.
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13.3
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Dot product;
properties of dot product
Angle between two vectors; Scalar projection; Work; Direction cosines
The
previous section showed you how to add vectors and stretch them (scalar
multiplication). This section gives one way to multiply vectors
with the result being a scalar. The dot product defined in this
section provides a convenient way to determine the angle between two vectors.
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13.4
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Cross product and
properties
Area of parallelogram: Scalar triple product; Volume of parallelepiped
This
section provides a second way to multiply two three-dimensional vectors - this
time producing a third vector perpendicular to the given vectors. Cross
product is defined only for three-dimensional vectors and has many uses; we
shall see, for example, in the next section that the cross product of two
vectors is useful for determining the equation of the plane formed by those
vectors.
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13.5
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Equations of lines in
space (vector, parametric, and symmetric: Parallel and skew lines
Normal vector; Equations of planes (vector and scalar): Parallel planes:
Angle between intersecting planes; Distance from a point to a plane and
distance between two planes
This
section gives three different, but equivalent "equations" for a
line in three-dimensional space - a vector form, a parametric form, and a
form similar to that of a line in two dimensions. Planes in
three-dimensional space, which may be thought of as all vectors perpendicular
to a given vector, will have an equation form very similar to that of lines
in two dimensions. We will use equations of lines and planes in later
chapters when we discuss tangent planes to surfaces - the analog of tangent
lines to graphs.
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13.6
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13.7
Surfaces
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14.1
Curves
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Vector Functions:
Limits: Continuity: Curves in Space
This
section extends the concepts of limits and continuity to vector
functions. The "graphs" of vector functions are curves in
two- or three-dimensional space. (In two dimensions, vector
functions are another way to look at curves defined by a pair of parametric
equations as was done in chapter 11.)
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14.2
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Derivatives of vector functions; Tangent vector; Properties of
derivatives
Smooth curves; Piecewise smooth curves
Integral of vector functions
This section
extends the concepts of derivatives and integrals to vector
functions. As you might expect, differentiation and integration
is performed component-wise.
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14.3
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Arc Length of an arc
in space: Arc length function: reparametrization
Curvature
Tangent vector: Unit normal vector: Normal plane
This
section show you how to determine the arc length of a smooth curve. Not
surprisingly, the process is the same as that for two-dimensional curves
defined paramaterically in chapter 11. This section also contains
several concepts for characterizing the shape of a curve; for example,
curvature - a measure of how fast the curve is bending at a given point.
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14.4
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Velocity; Speed;
Acceleration
Tangent and normal components of acceleration
All
previous notions, such as tangent vectors and curvature, are used in this
section to describe motion of an object in tree-dimensional space.
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Practice Test 1
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15.1
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Functions of Several
Variables
(AH) Sec 15.1 2, 5, 6, 14, 18, 30, 32, 40, 51-56
In
the last chapter we saw that the range of a function could be a
multi-dimensional vector. In this section we study functions
whose domains are multi-dimensional; that is, a function whose values are of
the form f (x, y) or f (x, y, z). (Later in
chapter 17 we sill study functions with both domain and range
multi-dimensional). A function of two variables, z = f(x, y),
will have a graph in three dimensions (x, y, and z).
Since these graphs are sometimes difficult to visualize, we introduce the
concept of a "level curve" - a subset of the domain that correspond
to a given functional value.
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15.2
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Limits and Continuity
(AH) Sec 15.2 4, 6, 12, 15, 24, 31, 37
This
section extends the concepts of limits and continuity to functions of two or
more variables. Be careful to note that a function can be "one
dimensionally" continuous along every line approaching a point and still
fail to be continuous at that point.
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15.3
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Partial Derivatives
(AH) Sec
15.3 4, 5, 6, 12, 20, 28, 35, 40, 50, 58, 64
This section starts the process of
finding derivatives for functions of two or more variables. For
two variables, we shall see that there is a derivative in the x-direction and
another in the y-direction and these may be obtained by a process similar to
that for functions of one variable. Each of these two
"partial" derivatives has, in turn, two partial derivatives; so the
original functions has four "second derivatives." Clairaut's
Theorem says that for some functions two of these four are the same.
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15.4
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Tangent Planes and
Linear Approximations
(AH) Sec 15.4 2, 6, 14, 17, 26, 30
This
section generalizes the notion of a tangent line for a curve to that of a
tangent plane for a surface. The tangent plane is determined by the
tangent lines in the x and y directions. Tangent
planes will be used to define a differential that may be used to find a
linear approximation of a functional value. Differentials and linear
approximations generalize to functions with more than two variables.
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15.5
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The Chain Rule
(AH) Sec 15.5 2, 6, 7, 10, 14, 21, 26, 34, 37
The
Chain Rule has different versions depending upon the number of variables
involved. The Chain Rule also comes in handy for finding derivatives
by implicit differentiation.
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15.6
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Directional
Derivatives and Gradient
(AH) Sec 15.6 2, 4, 9, 10, 14, 16, 23, 24, 34, 38, 42
The
partial derivatives fx (x, y) and fy (x,y) may be thought
of as the derivatives in the x-direction and y-directions,
respectively. This sections defines the "directional"
derivative for an arbitrary direction u=<a,b> in the xy-plane.
fx (x, y) and fy (x, y) will be special cases where the x-direction
is specified by i=<1,0> and the y-direction by j=<0,1>.
The vector that points in the direction where the directional derivatives is
the largest is called the gradient.
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15.7
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Maximum and Minimum
(AH) Sec 15.7 2, 6, 11, 16, 28, 31, 50
In
this section you will learn how to use partial derivatives to find the maxima
and minima of functions of two variables. The terms, concepts, and
procedures are very similar to those for functions of one variable.
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15.8
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LaGrange Multipliers
(AH) Sec 15.8 4, 8, 10, 18
Maximum
and minimum value problems are frequently stated in terms of finding the
maximum (or minimum) of a function given a constraint on the variables in the
form of an equation. This section gives another method for
doing some of the same type of problems done in Section 15.7.
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