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Advanced Engineering Mathematics

Erwin Kreyszig

Chapter 10

Vector Integral Calculus. Integral Theorems - all with Video Answers

Educators

Section 2

Peth Independence of Line Integrals

05:20

Problem 1

Show that the form under the integral sign is exact in the plane (Probs. $1-4$ ) or in space (Probs. $5-8$ ) and evaluate the integral. (Show the details of your work).
$$\int_{(0, \infty)}^{(1,-\infty)}(y \cos x y d x+x \cos x y d y)$$

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator
02:50

Problem 2

Show that the form under the integral sign is exact in the plane (Probs. $1-4$ ) or in space (Probs. $5-8$ ) and evaluate the integral. (Show the details of your work).
$$\int_{(\log 0)}^{(25)}\left(y^{2} e^{2 x} d x+y e^{2 x} d y\right)$$

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator
05:24

Problem 3

Show that the form under the integral sign is exact in the plane (Probs. $1-4$ ) or in space (Probs. $5-8$ ) and evaluate the integral. (Show the details of your work).
$$\int_{t-1,-1}^{(1, n)} e^{-x^{2}-y^{2}}(x d x+y d y)$$

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator
03:03

Problem 4

Show that the form under the integral sign is exact in the plane (Probs. $1-4$ ) or in space (Probs. $5-8$ ) and evaluate the integral. (Show the details of your work).
$$\int_{120}^{(2 \pi)}\left(\cos ^{2} y d x-2 x \cos y \sin y d y\right)$$

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator
04:15

Problem 5

Show that the form under the integral sign is exact in the plane (Probs. $1-4$ ) or in space (Probs. $5-8$ ) and evaluate the integral. (Show the details of your work).
$$\int_{0,3,0}^{(0,1)}\left(z e^{m} d x+d y+x e^{x x} d z\right)$$

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator
07:36

Problem 6

Show that the form under the integral sign is exact in the plane (Probs. $1-4$ ) or in space (Probs. $5-8$ ) and evaluate the integral. (Show the details of your work).
$$\int_{(a 0,0)}^{(1,3,0)} e^{-x^{2}+y^{2}-2 x}(x d x+y d y-d z)$$

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator
04:30

Problem 7

Show that the form under the integral sign is exact in the plane (Probs. $1-4$ ) or in space (Probs. $5-8$ ) and evaluate the integral. (Show the details of your work).
$$\int_{a, 0,00}^{\cos 0}\left(2 x y d x+x^{2} d y+\sinh z d z\right)$$

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator
04:41

Problem 8

Show that the form under the integral sign is exact in the plane (Probs. $1-4$ ) or in space (Probs. $5-8$ ) and evaluate the integral. (Show the details of your work).
$$\int_{(2,0,1)}^{(4,4)}\left[2 x\left(y^{2}-t^{3}\right) d x+3 x^{2} y^{2} d y-3 x^{2} z^{2} d z\right]$$

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator
02:26

Problem 9

Show that in Example 4 of the text we have $\mathbf{F}=\operatorname{grad}(\arctan (y / x)) \cdot$ Give examples of domains in which the integral is path independent.

R M
R M
Numerade Educator
05:03

Problem 10

(a) Show that $I=\int_{C}\left(x^{2} y d x+2 x y^{2} d y\right)$ is path dependent in the xy-plane
(b) Integrate from (0,0) along the straight-line segment to $(1, b), 0 \leq b \leq 1,$ and then vertically up to (1, 1): see the figure. For which of these paths is 1 maxirmum? What is its maximum value?
(c) Integrate from (0,0) along the straight-line segment to $(c, 1), 0 \leq c \leq 1,$ and then horizontally to $(1,1),$ For $c=1,$ do you get the same value as for $b=1$ in (b)? For which $c$ is $I$ maximum? What is its maximum value?

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator
02:14

Problem 11

and, if independent, integrate from (0,0,0) to $(a, b, c)$.
$$(\cosh x z)(z d x+x d z)$$

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator
01:26

Problem 12

and, if independent, integrate from (0,0,0) to $(a, b, c)$.
$$\left(3 x^{2} c^{2}-x\right) d x+2 x^{3} e^{2 x} d y$$

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator
02:09

Problem 13

and, if independent, integrate from (0,0,0) to $(a, b, c)$.
$$3 x^{2} y d x+x^{3} d y+y d z$$

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator
02:08

Problem 14

and, if independent, integrate from (0,0,0) to $(a, b, c)$.
$$2 x \sin y d x+x^{2} \cos y d y+y^{2} d z$$

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator
02:08

Problem 15

and, if independent, integrate from (0,0,0) to $(a, b, c)$.
$$\left(z e^{z}-e^{5}\right) d x-x e^{i x} d y+e^{z} d z$$

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator
02:17

Problem 16

and, if independent, integrate from (0,0,0) to $(a, b, c)$.
$$e^{x} \cos 2 y d x-2 e^{x} \sin 2 y d y-x z d x$$

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator
03:00

Problem 17

and, if independent, integrate from (0,0,0) to $(a, b, c)$.
$$x y z^{2} d x+\frac{1}{2} x^{2} z^{2} d y+x^{2} y z d z$$

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator
03:30

Problem 18

and, if independent, integrate from (0,0,0) to $(a, b, c)$.
$$y z \cosh x d x+z \sinh x d y+y \sinh x d z$$

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator
01:41

Problem 19

and, if independent, integrate from (0,0,0) to $(a, b, c)$.
$$y d x+(x-2 y) d y+4 x d z$$

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator
05:59

Problem 20

Make a list of the main ideas on path independence and dependence in this section. Then work this list into an essay. including explanations of all definitions and on the practical usefulaess of the theorems, but no proofs. Include illustrating examples of your own. Explain what happens in Example 4 if you take the domsin $0 < \sqrt{x^{2}+y^{2}} < \frac{3}{2}.$

Melissa Munoz
Melissa Munoz
Numerade Educator