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(a) Let div F⃗ =x2+y2+z2+1. Calculate ∫S1F⃗ ⋅dA⃗ where S1 is the sphere of radius 1 centered at the origin.
∫S1F⃗ ⋅dA⃗ =
Let S2 be the sphere of radius 4 centered at the origin; let S3 be the sphere of radius 6 centered at the origin; let S4 be the box of side 12 centered at the origin with edges parallel to the axes. Without calculating them, arrange the following integrals in increasing order:
A=∫S2F⃗ ⋅dA⃗ ,B=∫S3F⃗ ⋅dA⃗ ,C=∫S4F⃗ ⋅dA⃗ .
Use the divergence theorem to calculate the flux of the vector field F⃗ (x,y,z)=x4i⃗ +(3y−4x3y)j⃗ +2zk⃗ through the sphere S of radius 6 centered at the origin and oriented outward.
∬SF⃗ ⋅dA⃗ =
∬SF⃗ ⋅dA⃗ =
A vector field F⃗ has the property that the flux of F⃗ out of a small cube of side length 0.01 centered about the point (−1,−4,2) is 0.0025. Estimate div(F⃗ ) at the point (−1,−4,2).
div(F⃗ (−1,−4,2))≈
div(F⃗ (−1,−4,2))≈
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