 3.1: If a path x remains a constant distance from the origin, then the v...
 3.2: If a path is parametrized by arclength, then its velocity vector is...
 3.3: If a path is parametrized by arclength, then its velocity and accel...
 3.4: d dt x(t)=x (t).
 3.5: d dt (x y) = x dy dt + y dx dt
 3.6: = dT dt . 7.   = dB ds .
 3.7:   = dB ds .
 3.8: The curvature is always nonnegative
 3.9: The torsion is always nonnegative.
 3.10: N = dT ds .
 3.11: If a path x has zero curvature, then its acceleration is always par...
 3.12: If a path x has a constant binormal vectorB, then 0.
 3.13: d2s dt 2 2 + 2 ds dt 4 = a(t)2 .
 3.14: grad f is a scalar field.
 3.15: div F is a vector field.
 3.16: curl F is a vector field
 3.17: grad(div F) is a vector field.
 3.18: div(curl(grad f )) is a vector field.
 3.19: grad f div F is a vector field.
 3.20: The path x(t) = (2 cost, 4 sin t, t) is a flow line of the vector f...
 3.21: The path x(t) = (et cost, et (cost + sin t), et sin t) is a flow li...
 3.22: The vector field F = 2x y cosz i y2 cosz j + ex y k is incompressible
 3.23: The vector field F = 2x y cosz i y2 cosz j + ex y k is irrotational.
 3.24: ( f ) = 0 for all functions f : R3 R.
 3.25: If F = 0 and F = 0, then F = 0
 3.26: (F G) = F ( G) + G ( F).
 3.27: If F = curl G, then F is solenoidal.
 3.28: The vector field F = 2x sin y cosz i + x 2 cos y cosz j + x 2 sin y...
 3.29: There is a vector field F of class C2 on R3 such that F = x cos2 y ...
 3.30: If F and G are gradient fields, then F G is incompressible.
 3.31: A large piece of cylindrical metal pipe is to be manufactured to in...
 3.32: Suppose that x: I R3 is a path of class C3 parametrized by arclengt...
 3.33: Suppose that x: I R3 is a path of class C3 parametrized by arclengt...
 3.34: Suppose that x: I R3 is a path of class C3 parametrized by arclengt...
 3.35: Suppose that x: I R3 is a path of class C3 parametrized by arclengt...
 3.36: In this problem, we will find expressions for velocity and accelera...
 3.37: Suppose that the path x(t) = (sin 2t, 2 cos 2t,sin 2t 2) describes ...
 3.38: Suppose that the temperature at points inside a room is given by a ...
 3.39: Let F = u(x, y) i v(x, y) j be an incompressible, irrotational vect...
 3.40: Suppose that a particle of mass m travels along a path x according ...
 3.41: Let a particle of mass m travel along a differentiable path x in a ...
 3.42: Consider the situation in Exercise 41 and suppose that F is a centr...
 3.43: Can the vector field F = (ex cos y + ex sin z) i ex sin y j + ex co...
 3.44: Can the vector field F = x(y2 + 1) i + (yex ez ) j + x 2 ez k be th...
Solutions for Chapter 3: VectorValued Functions
Full solutions for Vector Calculus  4th Edition
ISBN: 9780321780652
Solutions for Chapter 3: VectorValued Functions
Get Full SolutionsThis textbook survival guide was created for the textbook: Vector Calculus, edition: 4. Chapter 3: VectorValued Functions includes 44 full stepbystep solutions. Vector Calculus was written by and is associated to the ISBN: 9780321780652. Since 44 problems in chapter 3: VectorValued Functions have been answered, more than 12457 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Acute triangle
A triangle in which all angles measure less than 90°

Annuity
A sequence of equal periodic payments.

Components of a vector
See Component form of a vector.

Decreasing on an interval
A function f is decreasing on an interval I if, for any two points in I, a positive change in x results in a negative change in ƒ(x)

Demand curve
p = g(x), where x represents demand and p represents price

Exponential form
An equation written with exponents instead of logarithms.

First octant
The points (x, y, z) in space with x > 0 y > 0, and z > 0.

Geometric sequence
A sequence {an}in which an = an1.r for every positive integer n ? 2. The nonzero number r is called the common ratio.

Initial side of an angle
See Angle.

Inverse cosine function
The function y = cos1 x

Law of sines
sin A a = sin B b = sin C c

Logarithmic reexpression of data
Transformation of a data set involving the natural logarithm: exponential regression, natural logarithmic regression, power regression

nth root of a complex number z
A complex number v such that vn = z

Piecewisedefined function
A function whose domain is divided into several parts with a different function rule applied to each part, p. 104.

Quadratic equation in x
An equation that can be written in the form ax 2 + bx + c = 01a ? 02

Shrink of factor c
A transformation of a graph obtained by multiplying all the xcoordinates (horizontal shrink) by the constant 1/c or all of the ycoordinates (vertical shrink) by the constant c, 0 < c < 1.

Sum of a finite arithmetic series
Sn = na a1 + a2 2 b = n 2 32a1 + 1n  12d4,

Transpose of a matrix
The matrix AT obtained by interchanging the rows and columns of A.

Vertices of an ellipse
The points where the ellipse intersects its focal axis.

yaxis
Usually the vertical coordinate line in a Cartesian coordinate system with positive direction up, pp. 12, 629.