Let F= r k r, where r = x i+ y j+ z k and k is a constant . (Note that if k = − 3 , this is an inverse-square field.) Let σ be the sphere of radius a centered at the origin and oriented by the outward normal n = r / r = r / a . (a) Find the flux of F across σ without performing any integration . (b) For what value of k is the flux independent of the radius of the sphere?
Let F= r k r, where r = x i+ y j+ z k and k is a constant . (Note that if k = − 3 , this is an inverse-square field.) Let σ be the sphere of radius a centered at the origin and oriented by the outward normal n = r / r = r / a . (a) Find the flux of F across σ without performing any integration . (b) For what value of k is the flux independent of the radius of the sphere?
Let
F=
r
k
r,
where r
=
x
i+
y
j+
z
k
and
k
is a constant
.
(Note that if
k
=
−
3
,
this is an inverse-square field.) Let
σ
be the sphere of radius a centered at the origin and oriented by the outward normal
n
=
r
/
r
=
r
/
a
.
(a) Find the flux of F across
σ
without performing any integration.
(b) For what value of kis the flux independent of the radius of the sphere?
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Let
F1 = (x + y) & + (-x+y)ŷ-2z2 and
F2 = 2yx+ (2x + 3z) ŷ +3yî.
Calculate the curl and divergence of F1 and F2. Which one can be written as the gradient
of a scalar field? Find a suitable potential that does the job.
Determine if each of the following vector fields is the gradient of a function f(x, y). If so, find all of the
functions with this gradient.
(a) (3x² + e¹0) i + (10x e¹0 - 9 siny) j
(b) (10x el0y 9 sin y) i + (3x² + e¹0y) j
a) I have placed my work and my answer on my answer sheet
For what values of b and c will F = (y2 + 2czx)i + y(bx + cz)j + ( y2 + cx2)k be a gradient field?
Thomas' Calculus: Early Transcendentals (14th Edition)
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