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Vectors and the Geometry of Space 11
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Vector Operations
3
Vector Operations
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In Theorem 11.3, u is called a unit vector in the direction of v. The process of multiplying v by to get a unit vector is called normalization of v.
Vector Operations
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The Dot Product
You have studied two operations with vectors—vector addition and multiplication by a scalar—each of which yields another vector. In this section you will study a third vector operation, called the dot product. This product yields a scalar, rather than a vector.
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The Dot Product
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Projections and Vector Components
The projection of u onto v can be written as a scalar multiple of a unit vector in the direction of v .That is,
The scalar k is called the component of u in the direction of v.
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The Cross Product
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A convenient way to calculate u × v is to use the following determinant form with cofactor expansion.
The Cross Product
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The Cross Product
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The Cross Product
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The Triple Scalar Product
For vectors u, v, and w in space, the dot product of u and v × w u (v × w) is called the triple scalar product, as defined in Theorem 11.9.
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If the vectors u, v, and w do not lie in the same plane, the triple scalar product u (v × w) can be used to determine the volume of the parallelepiped (a polyhedron, all of whose faces are parallelograms) with u, v, and w as adjacent edges, as shown in Figure 11.41.
Figure 11.41
The Triple Scalar Product
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Planes in Space
By regrouping terms, you obtain the general form of the equation of a plane in space.
Given the general form of the equation of a plane, it is easy to find a normal vector to the plane. Simply use the coefficients of x, y, and z and write n = .
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Distances Between Points, Planes, and Lines
If P is any point in the plane, you can find this distance by projecting the vector onto the normal vector n. The length of this projection is the desired distance.!
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Distances Between Points, Planes, and Lines
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Cylindrical Surfaces This circle is called a generating curve for the cylinder, as indicated in the following definition.
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Quadric Surfaces The fourth basic type of surface in space is a quadric surface. Quadric surfaces are the three-dimensional analogs of conic sections.
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Quadric Surfaces
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Quadric Surfaces cont’d
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Quadric Surfaces cont’d
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Surfaces of Revolution In a similar manner, you can obtain equations for surfaces of revolution for the other two axes, and the results are summarized as follows.
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The cylindrical coordinate system, is an extension of polar coordinates in the plane to three-dimensional space.
Cylindrical Coordinates
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Cylindrical to rectangular:
Rectangular to cylindrical:
The point (0, 0, 0) is called the pole. Moreover, because the representation of a point in the polar coordinate system is not unique, it follows that the representation in the cylindrical coordinate system is also not unique.
Cylindrical Coordinates
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Spherical Coordinates
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The relationship between rectangular and spherical coordinates is illustrated in Figure 11.75. To convert from one system to the other, use the following.
Spherical to rectangular:
Rectangular to spherical: Figure 11.75
Spherical Coordinates
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To change coordinates between the cylindrical and spherical systems, use the following.
Spherical to cylindrical (r ≥ 0):
Cylindrical to spherical (r ≥ 0):
Spherical Coordinates
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Vector-Valued Functions 12
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115
A
Space Curves and Vector-Valued Functions
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Limits and Continuity
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Differentiation of Vector-Valued Functions
The definition of the derivative of a vector-valued function parallels the definition given for real-valued functions.
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Differentiation of Vector-Valued Functions
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Differentiation of Vector-Valued Functions
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Integration of Vector-Valued Functions
The following definition is a rational consequence of the definition of the derivative of a vector-valued function.
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Velocity and Acceleration
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Tangent Vectors and Normal Vectors
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In Example 2, there are infinitely many vectors that are orthogonal to the tangent vector T(t). One of these is the vector T'(t) . This follows the property
T(t) T(t) = ||T(t)||2 =1 T(t) T'(t) = 0
By normalizing the vector T'(t) , you obtain a special vector called the principal unit normal vector, as indicated in the following definition.
Tangent Vectors and Normal Vectors
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The coefficients of T and N in the proof of Theorem 12.4 are called the tangential and normal components of acceleration and are denoted by aT = Dt [||v||] and aN = ||v|| ||T'||.
So, you can write
Tangential and Normal Components of Acceleration
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The following theorem gives some convenient formulas for aN and aT.
Tangential and Normal Components of Acceleration
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Arc Length
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Arc Length Parameter
Figure 12.30 210
Arc Length Parameter
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Functions of Several Variables 13
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For the function given by z = f(x, y), x and y are called the independent variables and z is called the dependent variable.
Functions of Several Variables
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Limit of a Function of Two Variables
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Continuity of a Function of Two Variables
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Continuity of a Function of Three Variables
A point (x0, y0, z0) in a region R in space is an interior point of R if there exists a δ-sphere about (x0, y0, z0) that lies entirely in R. If every point in R is an interior point, then R is called open.
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This definition indicates that if z = f(x, y), then to find fx you consider y constant and differentiate with respect to x.
Similarly, to find fy, you consider x constant and differentiate with respect to y.
Partial Derivatives of a Function of Two Variables
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Partial Derivatives of a Function of Two Variables
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Higher-Order Partial Derivatives
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Increments and Differentials
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Differentiability This is stated explicitly in the following definition.
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Differentiability
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Approximation by Differentials
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Chain Rules for Functions of Several Variables
Figure 13.39 367
The Chain Rule in Theorem 13.7 is shown schematically in Figure 13.41.
Figure 13.41
Chain Rules for Functions of Several Variables
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Implicit Partial Differentiation
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Directional Derivative
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The Gradient of a Function of Two Variables
The gradient of a function of two variables is a vector-valued function of two variables.
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The Gradient of a Function of Two Variables
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Applications of the Gradient
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Applications of the Gradient
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Functions of Three Variables
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Tangent Plane and Normal Line to a Surface
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A Comparison of the Gradients ∇f(x, y) and ∇F(x, y, z)
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Absolute Extrema and Relative Extrema
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Absolute Extrema and Relative Extrema
A minimum is also called an absolute minimum and a maximum is also called an absolute maximum. As in single-variable calculus, there is a distinction made between absolute extrema and relative extrema.
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Absolute Extrema and Relative Extrema
To locate relative extrema of f, you can investigate the points at which the gradient of f is 0 or the points at which one of the partial derivatives does not exist. Such points are called critical points of f.
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Absolute Extrema and Relative Extrema
It appears that such a point is a likely location of a relative extremum.
This is confirmed by Theorem 13.16.
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The Second Partials Test
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The Method of Least Squares
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If ∇f(x, y) = λ∇g(x, y) then scalar λ is called Lagrange multiplier.
Lagrange Multipliers
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Lagrange Multipliers
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Multiple Integration 14
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514
Area of a Plane Region
531
Double Integrals and Volume of a Solid Region
Using the limit of a Riemann sum to define volume is a special case of using the limit to define a double integral. The general case, however, does not require that the function be positive or continuous.
Having defined a double integral, you will see that a definite integral is occasionally referred to as a single integral.
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A double integral can be used to find the volume of a solid region that lies between the xy-plane and the surface given by z = f(x, y).
Double Integrals and Volume of a Solid Region
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Properties of Double Integrals Double integrals share many properties of single integrals.
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Evaluation of Double Integrals
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Average Value of a Function For a function f in one variable, the average value of f on [a, b] is
Given a function f in two variables, you can find the average value of f over the region R as shown in the following definition.
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This suggests the following theorem 14.3
Double Integrals in Polar Coordinates
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A lamina is assumed to have a constant density. But now you will extend the definition of the term lamina to include thin plates of variable density.
Double integrals can be used to find the mass of a lamina of variable density, where the density at (x, y) is given by the density function ρ.
Mass
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Surface Area
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14.5
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Moments and Center of Mass
By forming the Riemann sum of all such products and taking the limits as the norm of Δ approaches 0, you obtain the following definitions of moments of mass with respect to the x- and y-axes.
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Surface Area
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Taking the limit as leads to the following definition.
Triple Integrals
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Triple Integrals
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If f is a continuous function on the solid Q, you can write the triple integral of f over Q as
where the double integral over R is evaluated in polar coordinates. That is, R is a plane region that is either r-simple or θ-simple. If R is r-simple, the iterated form of the triple integral in cylindrical form is
Triple Integrals in Cylindrical Coordinates
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If (ρ, θ, φ) is a point in the interior of such a block, then the volume of the block can be approximated by ΔV ≈ ρ2 sin φ Δρ Δφ Δθ
Using the usual process involving an inner partition, summation, and a limit, you can develop the following version of a triple integral in spherical coordinates for a continuous function f defined on the solid region Q.
Triple Integrals in Spherical Coordinates
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In defining the Jacobian, it is convenient to use the following determinant notation.
Jacobians
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Change of Variables for Double Integrals Vector Analysis 15
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665
Vector Fields
Functions that assign a vector to a point in the plane or a point in space are called vector fields, and they are useful in representing various types of force fields and velocity fields.
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Note that an electric force field has the same form as a gravitational field. That is,
Such a force field is called an inverse square field.
Vector Fields
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Conservative Vector Fields
Some vector fields can be represented as the gradients of differentiable functions and some cannot—those that can are called conservative vector fields.
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Conservative Vector Fields
The following important theorem gives a necessary and sufficient condition for a vector field in the plane to be conservative.
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Curl of a Vector Field
The definition of the curl of a vector field in space is given below.
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Curl of a Vector Field
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Divergence of a Vector Field
You have seen that the curl of a vector field F is itself a vector field. Another important function defined on a vector field is divergence, which is a scalar function.
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Divergence of a Vector Field
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Line Integrals
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15.2
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Understand and use the concept of a piecewise smooth curve.
Write and evaluate a line integral.
Write and evaluate a line integral of a vector field.
Write and evaluate a line integral in differential form.
Objectives
709
Line Integrals
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Line Integrals
Note that if f(x, y, z) = 1, the line integral gives the arc length of the curve C. That is,
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Line Integrals of Vector Fields
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Fundamental Theorem of Line Integrals
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Independence of Path
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A curve C given by r(t) for a ≤ t ≤ b is closed if r(a) = r(b).
By the Fundamental Theorem of Line Integrals, you can conclude that if F is continuous and conservative on an open region R, then the line integral over every closed curve C is 0.
Independence of Path
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Green’s Theorem
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Among the many choices for M and N satisfying the stated condition, the choice of M = –y/2 and N = x/2 produces the following line integral for the area of region R.
Green’s Theorem
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Parametric Surfaces
807
Normal Vectors and Tangent Planes
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Area of a Parametric Surface
The area of the parallelogram in the tangent plane is
||∆uiru × ∆virv|| = ||ru × rv|| ∆ui∆vi
which leads to the following definition.
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Surface Integrals
853
Flux Integrals
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With these restrictions on S and Q, the Divergence Theorem is as follows.
Divergence Theorem
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Stokes’s Theorem
