HW4 - 1 Physics 2210 Homework assignment 4 B 6.6.5 ∇ φ =...

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Unformatted text preview: 1 Physics 2210. Homework assignment 4. B 6.6.5. ∇ φ = ∂φ ∂ x , ∂φ ∂ y , ∂φ ∂ z = − z,zcosy,siny - x ( ) . At point (2, π /2,-1), ∇ φ = ∂φ ∂ x , ∂φ ∂ y , ∂φ ∂ z = 1,0,-1 ( ) . At this point, the direction at which φ increases the fastest is n = ∇ φ | ∇ φ | = 2 2 ,0,- 2 2 . Therefore, the direction of fastest decreasing is m = − 2 2 ,0, 2 2 . To find the derivative of φ in the direction of 2i+3j, we first find the directional unit vector: s = 2 13 13 , 3 13 13 ,0 . Then d φ ds = ∇ φ ⋅ s = 2 13 13 . B 6.7.5. ∇⋅ V = ∂ V x ∂ x + ∂ V y ∂ y + ∂ V z ∂ z = 2x + 2y + 2z = 2(x + y + z) . ∇ × V = i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z V x V y V z = ( ∂ V z ∂ y- ∂ V y ∂ z ) i +( ∂ V x ∂ z- ∂ V z ∂ x ) j +( ∂ V y ∂ x- ∂ V x ∂ y ) k = 0 ....
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HW4 - 1 Physics 2210 Homework assignment 4 B 6.6.5 ∇ φ =...

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