 2.5.1: If f (x, y,z) = x 2 y3 + xyz, and x = 6t + 7, y = sin 2t, z = t 2, ...
 2.5.2: If f (x, y) = sin (x y) and x = s + t, y = s2 + t 2, find f/s and f...
 2.5.3: Suppose that a bird flies along the helical curve x = 2 cost, y = 2...
 2.5.4: Suppose that z = x 2 + y3, where x = st and y is a function of s an...
 2.5.5: You are the proud new owner of an Acme Deluxe Bread Kneading Machin...
 2.5.6: A rectangular stick of butter is placed in the microwave oven to me...
 2.5.7: Suppose that the following function is used to model the monthly de...
 2.5.8: The Centers for Disease Control and Prevention provides information...
 2.5.9: A cement mixer is pouring concrete in a conical pile. At the time w...
 2.5.10: A clarinetist is playing the glissando at the beginning of Rhapsody...
 2.5.11: Suppose z = f (x, y) has continuous partial derivatives. Let x = er...
 2.5.12: Suppose that z = f (x, y) has continuous partial derivatives. Let x...
 2.5.13: If w = g u2 v2, v2 u2 has continuous partial derivatives with respe...
 2.5.14: Suppose that z = f (x + y, x y) has continuous partial derivatives ...
 2.5.15: If w = f x y x 2 + y2 is a differentiable function of u = x y x 2 +...
 2.5.16: If w = f x 2 y2 x 2 + y2 is a differentiable function of u = x 2 y2...
 2.5.17: Suppose w = f y x x y , z x x z is a differentiable function of u =...
 2.5.18: Suppose that w = g x y , z y is a differentiable function of u = x/...
 2.5.19: In Exercises 1927, calculate D(f g) in two ways: (a) by first evalu...
 2.5.20: In Exercises 1927, calculate D(f g) in two ways: (a) by first evalu...
 2.5.21: In Exercises 1927, calculate D(f g) in two ways: (a) by first evalu...
 2.5.22: In Exercises 1927, calculate D(f g) in two ways: (a) by first evalu...
 2.5.23: In Exercises 1927, calculate D(f g) in two ways: (a) by first evalu...
 2.5.24: In Exercises 1927, calculate D(f g) in two ways: (a) by first evalu...
 2.5.25: In Exercises 1927, calculate D(f g) in two ways: (a) by first evalu...
 2.5.26: In Exercises 1927, calculate D(f g) in two ways: (a) by first evalu...
 2.5.27: In Exercises 1927, calculate D(f g) in two ways: (a) by first evalu...
 2.5.28: Let g: R3 R2 be a differentiable function such that g(1, 1, 3) = (2...
 2.5.29: Let g: R2 R2 and f: R2 R2 be differentiable functions such that g(0...
 2.5.30: Let z = f (x, y), where f has continuous partial derivatives. If we...
 2.5.31: (a) Use the methods of Example 6 and formula (10) in this section t...
 2.5.32: Show that the Laplacian operator 2/x 2 + 2/y2 + 2/z2 in three dimen...
 2.5.33: In this problem, you will determine the formula for the Laplacian o...
 2.5.34: Suppose that y is defined implicitly as a function y(x) by an equat...
 2.5.35: Find dy/dx when y is defined implicitly by the equation sin(x y) x ...
 2.5.36: Suppose that you are given an equation of the form F(x, y,z) = 0, f...
 2.5.37: Find z/x and z/y, where z is given implicitly by the equation x 3z ...
 2.5.38: Let f (x, y) = x 2 y x 2 + y2 if (x, y) = (0, 0) 0 if (x, y) = (0, ...
 2.5.39: Let w = f (x, y,z) be a differentiable function of x, y, and z. For...
 2.5.40: Let w = f (x, y,z) be a differentiable function of x, y, and z. For...
 2.5.41: Let w = f (x, y,z) be a differentiable function of x, y, and z. For...
 2.5.42: Let w = f (x, y,z) be a differentiable function of x, y, and z. For...
 2.5.43: Let w = f (x, y,z) be a differentiable function of x, y, and z. For...
 2.5.44: Let w = f (x, y,z) be a differentiable function of x, y, and z. For...
 2.5.45: Let w = f (x, y,z) be a differentiable function of x, y, and z. For...
Solutions for Chapter 2.5: The Chain Rule
Full solutions for Vector Calculus  4th Edition
ISBN: 9780321780652
Solutions for Chapter 2.5: The Chain Rule
Get Full SolutionsChapter 2.5: The Chain Rule includes 45 full stepbystep solutions. Vector Calculus was written by and is associated to the ISBN: 9780321780652. This textbook survival guide was created for the textbook: Vector Calculus, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Since 45 problems in chapter 2.5: The Chain Rule have been answered, more than 12722 students have viewed full stepbystep solutions from this chapter.

Acute angle
An angle whose measure is between 0° and 90°

Angle
Union of two rays with a common endpoint (the vertex). The beginning ray (the initial side) can be rotated about its endpoint to obtain the final position (the terminal side)

Augmented matrix
A matrix that represents a system of equations.

Future value of an annuity
The net amount of money returned from an annuity.

Horizontal component
See Component form of a vector.

Horizontal line
y = b.

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

Modified boxplot
A boxplot with the outliers removed.

Parabola
The graph of a quadratic function, or the set of points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

Probability of an event in a finite sample space of equally likely outcomes
The number of outcomes in the event divided by the number of outcomes in the sample space.

Pythagorean identities
sin2 u + cos2 u = 1, 1 + tan2 u = sec2 u, and 1 + cot2 u = csc2 u

Reflection across the xaxis
x, y and (x,y) are reflections of each other across the xaxis.

Repeated zeros
Zeros of multiplicity ? 2 (see Multiplicity).

Rose curve
A graph of a polar equation or r = a cos nu.

Sequence of partial sums
The sequence {Sn} , where Sn is the nth partial sum of the series, that is, the sum of the first n terms of the series.

Standard unit vectors
In the plane i = <1, 0> and j = <0,1>; in space i = <1,0,0>, j = <0,1,0> k = <0,0,1>

Sum identity
An identity involving a trigonometric function of u + v

Upper bound for ƒ
Any number B for which ƒ(x) ? B for all x in the domain of ƒ.

Vertical line test
A test for determining whether a graph is a function.

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