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Vector differentiation

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Question 1

What does the divergence of a vector field in vector calculus indicate?
 

  • The rotation of the vector field

  • The spreading or contracting of the vector field at a point

  • The integral of the vector field

  • The curl of the vector field

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Question 2

The divergence of the vector field [Tex]\vec{A} = x \hat{a}_x + y \hat{a}_y + z \hat{a}_z [/Tex] is

  • 0

  • 1/3

  • 1

  • 3

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Question 3

for the scalar field [Tex]u = {{{x^2}} \over 2} + {{{y^2}} \over 3},\,\,[/Tex] the magnitude of the gradient at the point [Tex](1,3)[/Tex] is

  • [Tex]\sqrt {{{13} \over 9}} [/Tex]

  • [Tex]\sqrt {{9 \over 2}} [/Tex]

  • [Tex]\sqrt 5 [/Tex]

  • [Tex]{{9 \over 2}}[/Tex]

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Question 4

What does the gradient of a scalar function represent?

  • The direction of steepest descent

  • The direction of steepest ascent

  • The second derivative of the function

  • The curvature of the function

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Question 5

For scalar functions f(x), g(x):


  • ∇(fg) = f∇g + g∇f

  • ∇(fg) = f∇g − g∇f

  • ∇(fg) = ∇f⋅∇g

  • ∇(fg) = ∇f +∇g

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Question 6

If[Tex] \overrightarrow{A} = \vec{r} = x\hat{i}+y\hat{j}+z\hat{k}[/Tex] then ∇([Tex]\overrightarrow{A}[/Tex] . [Tex]\overrightarrow{A}[/Tex]) = ?

  • 2 [Tex]\overrightarrow{r}[/Tex]

  • [Tex]\overrightarrow{r}[/Tex]

  • 0

  • [Tex]\overrightarrow{r}^{2}[/Tex]

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