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Questions and Answers
The divergence theorem of Gauss applies to a volume bounded by an open surface.
The divergence theorem of Gauss applies to a volume bounded by an open surface.
False (B)
A vector function of position involved in the divergence theorem must have continuous derivatives.
A vector function of position involved in the divergence theorem must have continuous derivatives.
True (A)
The normal vector in the divergence theorem is directed inward towards the volume.
The normal vector in the divergence theorem is directed inward towards the volume.
False (B)
One of the implications of Gauss's theorem is that it relates surface integrals to volume integrals.
One of the implications of Gauss's theorem is that it relates surface integrals to volume integrals.
The divergence theorem of Gauss is also known as Green's theorem.
The divergence theorem of Gauss is also known as Green's theorem.
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Study Notes
Gauss Divergence Theorem
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The theorem states that the surface integral of the normal component of a vector A taken over a closed surface is equal to the integral of the divergence of A taken over the volume enclosed by the surface.
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The formula for the theorem is:
∫∫(A • n̂)dS = ∫∫∫ (∇ • A)dV
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Where:
- A is a vector function of position with continuous derivatives
- n̂ is the positive (outward drawn) normal to the surface S
- V is the volume bounded by a closed surface S
- ∇ • A is the divergence of A
Applying the Theorem
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When applying the Gauss Divergence Theorem, remember to pay attention to signs and ensure that P is multiplied by x and Q is multiplied by y.
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The theorem can be applied to surfaces represented by z = f(x,y), x = g(y,z), or y = h(x,z), where f,g,h are single-valued, continuous, and differentiable.
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The position vector to any point of S is:
r = xi + yj + f(x,y)k
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The vector r is perpendicular to n, the normal vector.
Simplifying the Formula
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The formula can be further simplified to:
∫∫(A • n̂)dS = ∫∫F(x,y)dxdy
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Where R is the projection of S on the xy plane and F(x,y) is the value of A at a point on S.
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