Vector Fields and Divergence
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Questions and Answers

Is the divergence of the vector field P equal to 2XYZ + X?

  • Cannot be determined
  • False
  • The text does not provide enough information to answer the question
  • True (correct)

    • What is the divergence of the vector field P in cartesian coordinates?

  • 2XYZ + X (correct)
  • 2XYZ + Z
  • 2XYZ
  • 2XYZ - X

    • What is the divergence of the vector field P in cylindrical coordinates?

  • 2XYZ (correct)
  • 2XYZ + Z
  • 2XYZ + X
  • 2XYZ - X

    • What does Stoke's theorem state?

    <p>The circulation of a vector field A around a closed path I is equal to the surface integral of the curl of A over the open surface bounded by L.</p> Signup and view all the answers

    Is Stoke's theorem applicable only if A and △XA are continuous on S?

    <p>Yes</p> Signup and view all the answers

    What is the relationship between the circulation of a vector field A and the surface integral of the curl of A?

    <p>They are equal</p> Signup and view all the answers

    What is Stoke's theorem?

    <p>The circulation of a vector field $A$ around a closed path $I$ is equal to the surface integral of the curl of $A$ over the open surface bounded by $L$, provided $A$ and $\Delta XA$ are continuous on $S$.</p> Signup and view all the answers

    Applicability of Stoke's theorem

    <p>Yes, Stoke's theorem is applicable only if $A$ and $\Delta XA$ are continuous on $S$.</p> Signup and view all the answers

    Relationship between circulation and surface integral

    <p>The circulation of a vector field $A$ is equal to the surface integral of the curl of $A$.</p> Signup and view all the answers

    Study Notes

    Divergence of a Vector Field

    • The divergence of the vector field P is not equal to 2XYZ + X, as the expression lacks a differential operator.

    Divergence in Cartesian Coordinates

    • The divergence of the vector field P in Cartesian coordinates is given by ∇⋅P = (∂P_x/∂x) + (∂P_y/∂y) + (∂P_z/∂z).

    Divergence in Cylindrical Coordinates

    • The divergence of the vector field P in cylindrical coordinates is given by ∇⋅P = (1/r) (∂(rP_r)/∂r) + (1/r) (∂P_θ/∂θ) + (∂P_z/∂z).

    Stoke's Theorem

    • Stoke's theorem relates the circulation of a vector field around a closed curve to the surface integral of the curl of that vector field.

    Applicability of Stoke's Theorem

    • Stoke's theorem is applicable only if the vector field A and its first partial derivatives are continuous on the surface S and its boundary.

    Relationship between Circulation and Surface Integral

    • The circulation of a vector field A around a closed curve is equal to the surface integral of the curl of A, i.e., ∯A⋅dl = ∬ (∇×A) ⋅dS.

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    Description

    This quiz tests your knowledge of vector fields and divergence in Cartesian coordinates. The question asks whether the divergence of a given vector field is equal to a specific expression.

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