 8.2.1: Check that the parametrized 3manifold in Example 3 is in fact a sm...
 8.2.2: A planar robot arm is constructed by using two rods as shown in Fig...
 8.2.3: A planar robot arm is constructed by using a rod of length 3 anchor...
 8.2.4: A robot arm is constructed in R3 by anchoring a rod of length 2 to ...
 8.2.5: A robot arm is constructed in R3 by anchoring a rod of length 2 to ...
 8.2.6: Let a, b, and c be positive constants and x: [0, ] R3 the smooth pa...
 8.2.7: Evaluate C , where C is the unit circle x 2 + y2 = 1, oriented coun...
 8.2.8: Compute C , where C is the line segment in Rn from (0, 0,..., 0) to...
 8.2.9: Evaluate the integral X , where X is the parametrized helicoid X(s,...
 8.2.10: Consider the helicoid parametrized as X(u1, u2) = (u1 cos 3u2, u1 s...
 8.2.11: Let M be the subset of R3 given by {(x, y,z)  x 2 + y2 6 z 4 x 2 y...
 8.2.12: Calculate S , where S is the portion of the paraboloid z = x 2 + y2...
 8.2.13: Calculate S , where S is the portion of the cylinder x 2 + z2 = 4 w...
 8.2.14: Consider the parametrized 2manifold X: [1, 3] [0, 2) R4, X(s, t) =...
 8.2.15: Consider the parametrized 3manifold X: [0, 1] [0, 1] [0, 1] R4, X(...
Solutions for Chapter 8.2: Manifolds and Integrals of kforms
Full solutions for Vector Calculus  4th Edition
ISBN: 9780321780652
Solutions for Chapter 8.2: Manifolds and Integrals of kforms
Get Full SolutionsThis textbook survival guide was created for the textbook: Vector Calculus, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.2: Manifolds and Integrals of kforms includes 15 full stepbystep solutions. Vector Calculus was written by and is associated to the ISBN: 9780321780652. Since 15 problems in chapter 8.2: Manifolds and Integrals of kforms have been answered, more than 13491 students have viewed full stepbystep solutions from this chapter.

Algebraic expression
A combination of variables and constants involving addition, subtraction, multiplication, division, powers, and roots

Arccosecant function
See Inverse cosecant function.

Average velocity
The change in position divided by the change in time.

Base
See Exponential function, Logarithmic function, nth power of a.

Binomial coefficients
The numbers in Pascal’s triangle: nCr = anrb = n!r!1n  r2!

Circular functions
Trigonometric functions when applied to real numbers are circular functions

Degree of a polynomial (function)
The largest exponent on the variable in any of the terms of the polynomial (function)

Differentiable at x = a
ƒ'(a) exists

End behavior
The behavior of a graph of a function as.

Exponential decay function
Decay modeled by ƒ(x) = a ? bx, a > 0 with 0 < b < 1.

Gaussian elimination
A method of solving a system of n linear equations in n unknowns.

Higherdegree polynomial function
A polynomial function whose degree is ? 3

Horizontal asymptote
The line is a horizontal asymptote of the graph of a function ƒ if lim x: q ƒ(x) = or lim x: q ƒ(x) = b

Inverse sine function
The function y = sin1 x

Parametric equations for a line in space
The line through P0(x 0, y0, z 0) in the direction of the nonzero vector v = <a, b, c> has parametric equations x = x 0 + at, y = y 0 + bt, z = z0 + ct.

Product of complex numbers
(a + bi)(c + di) = (ac  bd) + (ad + bc)i

Radian
The measure of a central angle whose intercepted arc has a length equal to the circle’s radius.

Reference triangle
For an angle ? in standard position, a reference triangle is a triangle formed by the terminal side of angle ?, the xaxis, and a perpendicular dropped from a point on the terminal side to the xaxis. The angle in a reference triangle at the origin is the reference angle

RRAM
A Riemann sum approximation of the area under a curve ƒ(x) from x = a to x = b using x1 as the righthand end point of each subinterval.

yzplane
The points (0, y, z) in Cartesian space.