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.)
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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