Let σ be the surface of the solid G that is enclosed by the paraboloid z = 1 − x 2 − y 2 and the plane z = 0. Use a CAS to verify Formula (1) in the Divergence Theorem for the vector field F = x 2 y − z 2 i + y 3 − x j + 2 x + 3 z − 1 k by evaluating the surface integral and the triple integral.
Let σ be the surface of the solid G that is enclosed by the paraboloid z = 1 − x 2 − y 2 and the plane z = 0. Use a CAS to verify Formula (1) in the Divergence Theorem for the vector field F = x 2 y − z 2 i + y 3 − x j + 2 x + 3 z − 1 k by evaluating the surface integral and the triple integral.
Let
σ
be the surface of the solid G that is enclosed by the paraboloid
z
=
1
−
x
2
−
y
2
and the plane
z
=
0.
Use a CAS to verify Formula (1) in the Divergence Theorem for the vector field
F
=
x
2
y
−
z
2
i
+
y
3
−
x
j
+
2
x
+
3
z
−
1
k
by evaluating the surface integral and the triple integral.
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.
Show that V² (v-x) = 2V.v+x-V²v, where v is a vector field and x is the position
vector of a point in a 3D space.
Sketch the vector field of F
(-y, 0) in the xy-plane. Be sure to draw enough
vectors so that the pattern of the magnitude of the vectors and the direction of the
vectors is clear. It is not required to label any points or vectors.
Calculate the curl of the vector g
=
(x−y) ax+(y+x) ay, and visualize it. Explain
what does "curl of vector field" mean, in your own words. When does it equal
zero?
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