Math Problem Statement

prove that B=xyz^2 is solenoidal

Solution

To prove that the vector field B=xyz2i^+0j^+0k^\mathbf{B} = x y z^2 \hat{i} + 0 \hat{j} + 0 \hat{k} is solenoidal, we need to show that its divergence is zero.

Step-by-step proof:

  1. Definition of a solenoidal vector field: A vector field B\mathbf{B} is solenoidal if its divergence is zero everywhere, i.e., B=0\nabla \cdot \mathbf{B} = 0

  2. Given vector field: The vector field is B=xyz2i^+0j^+0k^\mathbf{B} = x y z^2 \hat{i} + 0 \hat{j} + 0 \hat{k}.

    In component form, we can express B\mathbf{B} as: B=(Bx,By,Bz)\mathbf{B} = \left( B_x, B_y, B_z \right) where: Bx=xyz2,By=0,Bz=0B_x = x y z^2, \quad B_y = 0, \quad B_z = 0

  3. Formula for the divergence: The divergence of a vector field B=(Bx,By,Bz)\mathbf{B} = (B_x, B_y, B_z) in Cartesian coordinates is: B=Bxx+Byy+Bzz\nabla \cdot \mathbf{B} = \frac{\partial B_x}{\partial x} + \frac{\partial B_y}{\partial y} + \frac{\partial B_z}{\partial z}

  4. Compute the partial derivatives:

    • For Bx=xyz2B_x = x y z^2: Bxx=x(xyz2)=yz2\frac{\partial B_x}{\partial x} = \frac{\partial}{\partial x}(x y z^2) = y z^2
    • For By=0B_y = 0: Byy=y(0)=0\frac{\partial B_y}{\partial y} = \frac{\partial}{\partial y}(0) = 0
    • For Bz=0B_z = 0: Bzz=z(0)=0\frac{\partial B_z}{\partial z} = \frac{\partial}{\partial z}(0) = 0
  5. Substitute into the divergence formula: B=yz2+0+0=yz2\nabla \cdot \mathbf{B} = y z^2 + 0 + 0 = y z^2

  6. Conclusion: The divergence of B\mathbf{B} is: B=yz2\nabla \cdot \mathbf{B} = y z^2 Since B0\nabla \cdot \mathbf{B} \neq 0, the vector field B=xyz2i^+0j^+0k^\mathbf{B} = x y z^2 \hat{i} + 0 \hat{j} + 0 \hat{k} is not solenoidal.

Therefore, B\mathbf{B} is not solenoidal, as its divergence is not zero.


Do you want further details or have any questions?

Here are 5 related questions to deepen your understanding:

  1. What are the physical implications of a vector field being solenoidal?
  2. How is the solenoidal condition used in fluid dynamics?
  3. Can a vector field that is not solenoidal be transformed into a solenoidal one?
  4. How does the divergence theorem relate to solenoidal fields?
  5. What are examples of solenoidal vector fields in electromagnetism?

Tip: The divergence of a vector field helps measure how much the field spreads out from a point. Solenoidal fields represent incompressible flows, where there is no net outflow from any point.

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Math Problem Analysis

Mathematical Concepts

Vector Fields
Divergence

Formulas

Divergence Formula in Cartesian Coordinates

Theorems

Solenoidal Vector Fields

Suitable Grade Level

College Level