Use Green’s Theorem in the form of Equation 13 to prove Green’s first identity: ∬ D f ∇ 2 g d A = ∮ c f ( ∇ g ) ⋅ n d s − ∬ D ∇ f ⋅ ∇ g d A Where D and C satisfy the hypotheses of Green’s Theorem and the appropriate partial derivatives of f and g exist and are continuous. (The quantity ∇ g ⋅ n = D n g occurs in the line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of g .)
Use Green’s Theorem in the form of Equation 13 to prove Green’s first identity: ∬ D f ∇ 2 g d A = ∮ c f ( ∇ g ) ⋅ n d s − ∬ D ∇ f ⋅ ∇ g d A Where D and C satisfy the hypotheses of Green’s Theorem and the appropriate partial derivatives of f and g exist and are continuous. (The quantity ∇ g ⋅ n = D n g occurs in the line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of g .)
Solution Summary: The author explains that Green's first identity is undersetDoverset'iint, where D and C satisfy the hypotheses of Green’s Theorem.
Use Green’s Theorem in the form of Equation 13 to prove Green’s first identity:
∬
D
f
∇
2
g
d
A
=
∮
c
f
(
∇
g
)
⋅
n
d
s
−
∬
D
∇
f
⋅
∇
g
d
A
Where D and C satisfy the hypotheses of Green’s Theorem and the appropriate partial derivatives of f and g exist and are continuous. (The quantity
∇
g
⋅
n
=
D
n
g
occurs in the line integral. This is the directional derivative in the direction of the normal vectorn and is called the normal derivative of g.)
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.
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