(a) Let D = R² \ {(0,0)} and consider the vector field f defined by: f(x, y) = f(x, y)i + f2(x, y) j where X x² + y² x³ x²y = + +2xy cos(x²y) and f2(x, y) 6 2 fi(x, y) Show that f is a gradient field by finding a potential function a scalar function o such that Vo = f. + 23 + Y x² + y² 6 for it; i.e, find -+x² cos(x-
(a) Let D = R² \ {(0,0)} and consider the vector field f defined by: f(x, y) = f(x, y)i + f2(x, y) j where X x² + y² x³ x²y = + +2xy cos(x²y) and f2(x, y) 6 2 fi(x, y) Show that f is a gradient field by finding a potential function a scalar function o such that Vo = f. + 23 + Y x² + y² 6 for it; i.e, find -+x² cos(x-
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:(a) Let D = R² \ {(0, 0)} and consider the vector field f defined by:
f(x, y) = f₁(x, y)i + f2(x, y) j
where
X
fi(x, y) = 2² + y² +²
x3
x²y
+ +2xy cos(x²y) and f₂(x, y)
6 2
=
Y
x² + y²
x3
+
-+x² cos(x²y)
6
Show that f is a gradient field by finding a potential function for it; i.e, find
a scalar function o such that Vo = f.
y
4
(b) Let C be the curve defined by the intersection of the paraboloid z = x² - +
and the plane y = 2, whose starting and end points are (0, 2, 1) and (2, 2, 5).
Let f be a scalar function defined by f(x, y) = 2x³ + 2xz + xy.
Compute the integral: fds, where s is the arc length parametrisation of
C.
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