 2.1: In the following exercises you may assume that the exponential, sin...
 2.2: In the following exercises you may assume that the exponential, sin...
 2.3: Compute the limits: (a) limit (x, y) (0,1) x3 y (b) limit x0 cos x ...
 2.4: Compute the following limits: (a) limit (x, y) (0,1) ex y (b) limit...
 2.5: Compute the following limits: (a) limit x3 (x2 3x + 5) (b) limit x0...
 2.6: Let f (x, y) = xy3 x2+y6 if (x, y) = (0, 0) 0 if (x, y) = (0, 0). (...
 2.7: Let f (x, y, z) = ex+y 1 + z2 . Compute limh0 f (1,2+h,3) f (1,2,3)...
 2.8: Compute the following limits if they exist: (a) limit (x, y) (0,0) ...
 2.9: Compute the following limits if they exist: (a) limit (x, y) (0,0) ...
 2.10: Compute the following limits if they exist: (a) limit (x, y) (0,0) ...
 2.11: Compute the following limits if they exist: (a) limit (x, y) (0,0) ...
 2.12: Compute the following limits if they exist: (a) limit x 0 sin 2x 2x...
 2.13: Compute limitxx0 f (x), if it exists, for the following cases: (a) ...
 2.14: Let f (x, y, z) = 1 x2+y2+z21 . Describe geometrically the set in R...
 2.15: Where is the function f (x, y) = 1 x2+y2 continuous?
 2.16: Let A = 1 2 3 4 . (a) Considering A: R2 R2 as a linear map, explici...
 2.17: Find lim (x, y)(0,0)(3x2 + 3y2) log(x2 + y2). (HINT: Use polar coor...
 2.18: Show that the subsets of the plane in Exercises 1821 are open: A = ...
 2.19: Show that the subsets of the plane in Exercises 1821 are open: B = ...
 2.20: Show that the subsets of the plane in Exercises 1821 are open: C = ...
 2.21: Show that the subsets of the plane in Exercises 1821 are open: D = ...
 2.22: Let A R2 be the open unit disk D1(0, 0) with the point x0 = (1, 0) ...
 2.23: If f : Rn R and g: Rn R are continuous, show that the functions f 2...
 2.24: (a) Show that f : R R, x (1 x)8 + cos (1 + x3) is continuous. (b) S...
 2.25: (a) Can [sin (x + y)]/(x + y) be made continuous by suitably defini...
 2.26: Using either s and s or spherical coordinates, show that limit (x, ...
 2.27: Use the  formulation of limits to prove that x2 4 as x 2. Give ano...
 2.28: (a) Prove that for x Rn and s < t, Ds(x) Dt(x). (b) Prove that if U...
 2.29: Suppose x and y are in Rn and x = y. Show that there is a continuou...
 2.30: Let f : A Rn R be given and let x0 be a boundary point of A. We say...
 2.31: Let b R and f : R\[b] R be a function. We write limitxb f (x) = L a...
 2.32: Show that f is continuous at x0 if and only if limit xx0 f (x) f (x...
 2.33: Let f : A Rn Rm satisfy f (x) f (y) K x y for all x and y in A fopo...
 2.34: Show that f : Rn Rm is continuous at all points if and only if the ...
 2.35: (a) Find a specific number > 0 such that if a < , then a3 + 3a2 ...
Solutions for Chapter 2: Differentiation
Full solutions for Vector Calculus  6th Edition
ISBN: 9781429215084
Solutions for Chapter 2: Differentiation
Get Full SolutionsVector Calculus was written by and is associated to the ISBN: 9781429215084. This textbook survival guide was created for the textbook: Vector Calculus, edition: 6. Chapter 2: Differentiation includes 35 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 35 problems in chapter 2: Differentiation have been answered, more than 3030 students have viewed full stepbystep solutions from this chapter.

Associative properties
a + (b + c) = (a + b) + c, a(bc) = (ab)c.

Combinatorics
A branch of mathematics related to determining the number of elements of a set or the number of ways objects can be arranged or combined

Complements or complementary angles
Two angles of positive measure whose sum is 90°

Compounded continuously
Interest compounded using the formula A = Pert

Cotangent
The function y = cot x

Cube root
nth root, where n = 3 (see Principal nth root),

Division
a b = aa 1 b b, b Z 0

Horizontal translation
A shift of a graph to the left or right.

Implied domain
The domain of a function’s algebraic expression.

Multiplication property of equality
If u = v and w = z, then uw = vz

Multiplicative inverse of a real number
The reciprocal of b, or 1/b, b Z 0

Nonsingular matrix
A square matrix with nonzero determinant

Probability function
A function P that assigns a real number to each outcome O in a sample space satisfying: 0 … P1O2 … 1, P12 = 0, and the sum of the probabilities of all outcomes is 1.

Quotient identities
tan ?= sin ?cos ?and cot ?= cos ? sin ?

Reexpression of data
A transformation of a data set.

Real zeros
Zeros of a function that are real numbers.

Relevant domain
The portion of the domain applicable to the situation being modeled.

Rigid transformation
A transformation that leaves the basic shape of a graph unchanged.

Solve by substitution
Method for solving systems of linear equations.

Vector equation for a line in space
The line through P0(x 0, y0, z0) in the direction of the nonzero vector V = <a, b, c> has vector equation r = r0 + tv , where r = <x,y,z>.