(1.) (a.) Show that the field F i VXF= = (3x²y³ + zez, 3x³y² + 1±², tan¯¹(y) + xe**) is conservative. d 02 1+ y² 1+Y , xZ 3XY+Ze 3x + tàn ( xẻ Ха 2 Since I is cont он (b.) Construct its scalar potential function f so that F = Vf XZ f = S3x²y³ + Ze z dx + 3 (Y, Z) 3.3 xz ху+е +9(Y,Z) 32 Fy = 3x y + Jy (y, z) -1 = x= e +xze xz - (ex²+xze²²) 9x²y 2 -9x²4² > (0,0,0) Cous XZ fe xe +ta (y) +\'(z) =)h(z)=0 h(z)=c =>gy (4,2) = => g = ztaly) + h (z) 1+42 >> f = x ³ y ³ -1 33 xz ztan (y) + h(z) = xy + e + 31 XZ f(x,y,z) - x 34³ + (x² + ztan (x) e +C F. dr where C is the piecewise smooth curve that travels from (3,0,0) to (0,2,0) along = 1 and then along the line segment to the point (0, 0, 1). (c.) Determine x2 the ellipse y² + 9 4 (c=U) USE FTLI: 33 0.1 3.0 f(0,0,1) -f (3,0,0) = (00² + e + 1·m (1)) - (3+e" + 0.th" (0) = (0+1+1) - (0+1+0) =141-1
(1.) (a.) Show that the field F i VXF= = (3x²y³ + zez, 3x³y² + 1±², tan¯¹(y) + xe**) is conservative. d 02 1+ y² 1+Y , xZ 3XY+Ze 3x + tàn ( xẻ Ха 2 Since I is cont он (b.) Construct its scalar potential function f so that F = Vf XZ f = S3x²y³ + Ze z dx + 3 (Y, Z) 3.3 xz ху+е +9(Y,Z) 32 Fy = 3x y + Jy (y, z) -1 = x= e +xze xz - (ex²+xze²²) 9x²y 2 -9x²4² > (0,0,0) Cous XZ fe xe +ta (y) +\'(z) =)h(z)=0 h(z)=c =>gy (4,2) = => g = ztaly) + h (z) 1+42 >> f = x ³ y ³ -1 33 xz ztan (y) + h(z) = xy + e + 31 XZ f(x,y,z) - x 34³ + (x² + ztan (x) e +C F. dr where C is the piecewise smooth curve that travels from (3,0,0) to (0,2,0) along = 1 and then along the line segment to the point (0, 0, 1). (c.) Determine x2 the ellipse y² + 9 4 (c=U) USE FTLI: 33 0.1 3.0 f(0,0,1) -f (3,0,0) = (00² + e + 1·m (1)) - (3+e" + 0.th" (0) = (0+1+1) - (0+1+0) =141-1
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Please review and solve the following problem. The screenshot listed already has some work done and the correct answer listed. Please solve the problem and include an explanation of how the work was solved. Also, Please make sure to double check the answer provided matches up with the screenshot and the work is properly formatted so I am able to follow along. Thanks :)
Transcribed Image Text:(1.) (a.) Show that the field F
i
VXF=
=
(3x²y³ + zez, 3x³y² + 1±², tan¯¹(y) + xe**) is conservative.
d
02
1+ y²
1+Y
,
xZ
3XY+Ze
3x
+
tàn ( xẻ
Ха
2
Since I is
cont
он
(b.) Construct its scalar potential function f so that F = Vf
XZ
f = S3x²y³ + Ze z dx + 3 (Y, Z)
3.3 xz
ху+е +9(Y,Z)
32
Fy = 3x y + Jy (y, z)
-1
=
x=
e +xze
xz
-
(ex²+xze²²)
9x²y 2 -9x²4² >
(0,0,0)
Cous
XZ
fe xe
+ta (y) +\'(z)
=)h(z)=0
h(z)=c
=>gy (4,2) =
=> g = ztaly) + h (z)
1+42
>> f = x ³ y ³
-1
33 xz ztan (y) + h(z)
= xy + e +
31
XZ
f(x,y,z) - x 34³ + (x² + ztan (x)
e
+C
F. dr where C is the piecewise smooth curve that travels from (3,0,0) to (0,2,0) along
= 1 and then along the line segment to the point (0, 0, 1).
(c.) Determine
x2
the ellipse
y²
+
9
4
(c=U)
USE FTLI:
33
0.1
3.0
f(0,0,1) -f (3,0,0) = (00² + e + 1·m (1)) - (3+e" + 0.th" (0)
= (0+1+1) - (0+1+0)
=141-1
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