In parts (a) - (h), prove the property for vector fields F and G and scalar function f . (Assume that the required partial derivatives are continuous.) a) curl( F + G = curlF + curlG b) c u r l ( ∇ f ) = ∇ x ( ∇ f ) = 0 c) div( ∇ f ) = ∇ x ( ∇ f ) = 0 d) div(FxG ) = ( curl G − F ( curlG ) e) ∇ x [ ∇ f + ( ∇ x F ) ] = ∇ x ( ∇ x F ) f) ∇ x ( f F ) = f ( ∇ x F ) + ( ∇ f ) x F g) d i v ( f F ) = f ( d i v F ) + ( ∇ f ) . F h) d i v ( c u r l F ) = 0
In parts (a) - (h), prove the property for vector fields F and G and scalar function f . (Assume that the required partial derivatives are continuous.) a) curl( F + G = curlF + curlG b) c u r l ( ∇ f ) = ∇ x ( ∇ f ) = 0 c) div( ∇ f ) = ∇ x ( ∇ f ) = 0 d) div(FxG ) = ( curl G − F ( curlG ) e) ∇ x [ ∇ f + ( ∇ x F ) ] = ∇ x ( ∇ x F ) f) ∇ x ( f F ) = f ( ∇ x F ) + ( ∇ f ) x F g) d i v ( f F ) = f ( d i v F ) + ( ∇ f ) . F h) d i v ( c u r l F ) = 0
In parts (a) - (h), prove the property for vector fields F and G and scalar function f. (Assume that the required partial derivatives are continuous.)
a)
curl(
F
+
G
=
curlF
+
curlG
b)
c
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(
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=
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=
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c)
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=
∇
x
(
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f
)
=
0
d)
div(FxG
)
=
(
curl
G
−
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(
curlG
)
e)
∇
x
[
∇
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+
(
∇
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F
)
]
=
∇
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(
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x
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∇
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f
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)
=
f
(
∇
x
F
)
+
(
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)
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F
g)
d
i
v
(
f
F
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=
f
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d
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+
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F
h)
d
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v
(
c
u
r
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F
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=
0
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.
7. (a) Determine the value of k for which (u, v, w) is an orthogonal coordinate system ifx = −(u2 + kv2), y = uv and z = w.(b) Given that F and G are vector fields with G a vector potential of F, prove that G is notunique.(c) Show that a vector field F =(4uv − θ3/√u2 + v2, 2u2/√u2 + v2, (lnθ − 3uθ2/uv), defined in paraboloidalcoordinate system (x = uv cos θ, y = uv sin θ, z =1/2(u2 − v2) is irrotational and hence find its scalar potential.
Sketch the vector field F(x,y)=1/2xi-1/2yj at points (-1,1), (-1,0),(-1,1),(0,-1),(0,0),(0,1),(1,-1),(1,0),(1,1)
Which of the following statements are true for all vector fields, and which are true only for conservative vector fields? (a) The line integral along a path from P to Q does not depend on which path is chosen. (b) The line integral over an oriented curve C does not depend on how C is parametrized. (c) The line integral around a closed curve is zero. (d) The line integral changes sign if the orientation is reversed. (e ) The line integral is equal to the difference of a potential function at the two endpoints. (f) The line integral is equal to the integral of the tangential component along the curve. (g) The cross partial derivatives of the components are equal. 3
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