 16.16.1: The accompanying figore shows two polygonal paths in space joining ...
 16.16.2: The accompanying figure shows three polygonal paths joining the ori...
 16.16.3: Integrate f(x,y, z) = ~ over the circle r(t) = (a cos t)j + (a sin ...
 16.16.4: Integrate f(x, y, z) = v' x 2 + y2 over the involute curve r(t) = (...
 16.16.5: Evaluate the integrals in Exercises 5 and 6.
 16.16.6: Evaluate the integrals in Exercises 5 and 6.
 16.16.7: Integrate F = (ysinz)i + (xsinz)j + (xycosz)k around the circle cu...
 16.16.8: Integrate F = 3x2yi + (x 3 + l)j + 9z2k around the circle cut from ...
 16.16.9: Evaluate the integrals in Exercises 9 and 10. 1sxsinydx  SycosxdyC...
 16.16.10: Evaluate the integrals in Exercises 9 and 10. 1y2 dx + x 2 dyC is t...
 16.16.11: Find the area of the elliptical region cut from the plane x + y + z...
 16.16.12: Find the area of the cap cut from the paraboloid y2 + z2 = 3x by th...
 16.16.13: Find the area of the cap cut from the top of the sphere x 2 + y2 + ...
 16.16.14: a. Hemisphere cut by cylinder Find the area of the surface cut from...
 16.16.15: Find the area of the triangle in which the plane (x/a) + (y/b) + (z...
 16.16.16: Integrate yz a. g(x,y,z) = ~r /~= v4y2 + 1 b. g(x,y,z) = v'4Y~ + 1...
 16.16.17: Integrate g(x, y, z) = x 4y(y2 + z2) over the portion of the cylind...
 16.16.18: The state of Wyoming is bounded by the meridians 111 3 / and 1043/ ...
 16.16.19: Find parametrizations for the surfaces in Exercises 19 24. (There ...
 16.16.20: Find parametrizations for the surfaces in Exercises 19 24. (There ...
 16.16.21: Find parametrizations for the surfaces in Exercises 19 24. (There ...
 16.16.22: Find parametrizations for the surfaces in Exercises 19 24. (There ...
 16.16.23: Find parametrizations for the surfaces in Exercises 19 24. (There ...
 16.16.24: Find parametrizations for the surfaces in Exercises 19 24. (There ...
 16.16.25: The portion of the paraboloid y = 2(x2 + z2), y :5 2, that lies abo...
 16.16.26: Integrate /(x,y,z) = xy  z2 over the sur face in Exercise 25
 16.16.27: Find the surface area of the helicoid r(r, 6) = (r cos 6)i + (rsin ...
 16.16.28: Evaluate the integral ffs v'x2 + y2 + I du, wbere S is the helicoid...
 16.16.29: Which of the fields in Exercises 2932 are conservative, and which ...
 16.16.30: Which of the fields in Exercises 2932 are conservative, and which ...
 16.16.31: Which of the fields in Exercises 2932 are conservative, and which ...
 16.16.32: Which of the fields in Exercises 2932 are conservative, and which ...
 16.16.33: Find potential functions for the fields in Exercises 33 and 34. F =...
 16.16.34: Find potential functions for the fields in Exercises 33 and 34. F =...
 16.16.35: In Exercises 35 and 36, Imd the work done by each field along the p...
 16.16.36: In Exercises 35 and 36, Imd the work done by each field along the p...
 16.16.37: Find the work done by xi + yj F = C;;C,,= (x2 + y2)'/2 over th...
 16.16.38: Find the flow of the field F = V(x2 ze') a. once around the ellipse...
 16.16.39: In Exercises 39 and 40, use the surface integral in Stokes' Theorem...
 16.16.40: In Exercises 39 and 40, use the surface integral in Stokes' Theorem...
 16.16.41: Find the mass of a thin wire lying along the curve r(l) = v2ti + v2...
 16.16.42: Find the center of mass of a thin wire lying along the curve r(l) =...
 16.16.43: Find the ceoter of mass and the momeots of inertia aboot the coordi...
 16.16.44: A slender metal arch lies along the semicircle y = v' a2  x 2 in t...
 16.16.45: A wire of constant density B = 1 lies along the curve r(l) = (e' co...
 16.16.46: Find the mass and center of mass of a wire of constant density B th...
 16.16.47: Find I. and the center of mass of a thin shell of density B(x, y, z...
 16.16.48: Find the moment of inertia about the zaxis of the surface of the c...
 16.16.49: Use Green's Theorem to Imd the counterclockwise circulation and oot...
 16.16.50: Use Green's Theorem to Imd the counterclockwise circulation and oot...
 16.16.51: Show that f cosy lnxsinyt(y  xtix ~ 0 c for any closed curve C t...
 16.16.52: Show that tire outward flux of the position vector field F = xi + y...
 16.16.53: In Exercises 5356, fmd tire outward flux of F across tire bouodary...
 16.16.54: In Exercises 5356, fmd tire outward flux of F across tire bouodary...
 16.16.55: In Exercises 5356, fmd tire outward flux of F across tire bouodary...
 16.16.56: In Exercises 5356, fmd tire outward flux of F across tire bouodary...
 16.16.57: Let S he the surface that is bounded on tire leftby the hemisphere ...
 16.16.58: Find the outward flux of the fteld F ~ 3X2'i + yj  z'k across the ...
 16.16.59: Use the Divergence Theorem to fmd the flux ofF ~ xy'i + x'yj + yk o...
 16.16.60: Find the flux ofF ~ (3z + I)kupwardacrossthe hemisphere x 2 + y2 + ...
Solutions for Chapter 16: Integration in Vector Fields
Full solutions for Thomas' Calculus  12th Edition
ISBN: 9780321587992
Solutions for Chapter 16: Integration in Vector Fields
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 60 problems in chapter 16: Integration in Vector Fields have been answered, more than 11551 students have viewed full stepbystep solutions from this chapter. Thomas' Calculus was written by and is associated to the ISBN: 9780321587992. This textbook survival guide was created for the textbook: Thomas' Calculus, edition: 12. Chapter 16: Integration in Vector Fields includes 60 full stepbystep solutions.

Additive inverse of a real number
The opposite of b , or b

Control
The principle of experimental design that makes it possible to rule out other factors when making inferences about a particular explanatory variable

Data
Facts collected for statistical purposes (singular form is datum)

Deductive reasoning
The process of utilizing general information to prove a specific hypothesis

Frequency distribution
See Frequency table.

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.

Intermediate Value Theorem
If ƒ is a polynomial function and a < b , then ƒ assumes every value between ƒ(a) and ƒ(b).

Law of cosines
a2 = b2 + c2  2bc cos A, b2 = a2 + c2  2ac cos B, c2 = a2 + b2  2ab cos C

Linear regression line
The line for which the sum of the squares of the residuals is the smallest possible

LRAM
A Riemann sum approximation of the area under a curve ƒ(x) from x = a to x = b using x1 as the lefthand endpoint of each subinterval

Power regression
A procedure for fitting a curve y = a . x b to a set of data.

Probability simulation
A numerical simulation of a probability experiment in which assigned numbers appear with the same probabilities as the outcomes of the experiment.

Quadratic function
A function that can be written in the form ƒ(x) = ax 2 + bx + c, where a, b, and c are real numbers, and a ? 0.

Regression model
An equation found by regression and which can be used to predict unknown values.

Row operations
See Elementary row operations.

Sine
The function y = sin x.

Standard deviation
A measure of how a data set is spread

Sum of functions
(ƒ + g)(x) = ƒ(x) + g(x)

Symmetric difference quotient of ƒ at a
ƒ(x + h)  ƒ(x  h) 2h

Zero vector
The vector <0,0> or <0,0,0>.