\( \newcommand{\sech}{\operatorname{sech}} \) \( \newcommand{\inverse}[1]{#1^\leftarrow} \) \( \newcommand{\<}{\langle} \) \( \newcommand{\>}{\rangle} \) \( \newcommand{\vect}{\mathbf} \) \( \newcommand{\veci}{\mathbf{\hat ı}} \) \( \newcommand{\vecj}{\mathbf{\hat ȷ}} \) \( \newcommand{\veck}{\mathbf{\hat k}} \) \( \newcommand{\curl}{\operatorname{curl}\,} \) \( \newcommand{\dv}{\operatorname{div}\,} \) \( \newcommand{\detThree}[9]{ \operatorname{det}\left( \begin{array}{c c c} #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right) } \) \( \newcommand{\detTwo}[4]{ \operatorname{det}\left( \begin{array}{c c} #1 & #2 \\ #3 & #4 \end{array} \right) } \)

MA 227 Standard C10

Vector Fields
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At the end of the course, each student should be able to…

  • C10: VectField. Analyze vector fields, including computing curl and divergence.

C10. Vector Fields

  • A vector field \(\vect F=\<M,N,P\>\) is a function of \(n\) variables that yields a vector of dimension \(n\).
    • Vector fields are sketched by choosing several points \(P\) and plotting the vectors \(\vect F(P)\) starting from each point.
    • The gradient vector field \(\nabla f\) represents the change of the function \(f\) at each point. If \(f\) is temperature, then \(\nabla f\) is heat flux.
  • The nabla operator \(\nabla=\<\frac{\partial}{\partial x}, \frac{\partial}{\partial y},\frac{\partial}{\partial z}\>\) gives a convenient shorthand for computing certain derivative operations.
    • The gradient vector field is the scalar product of \(\nabla\) with the function \(f\).
  • The rotations of a vector field are given by curl.
    • Each component of \(\curl \vect F = \nabla\times\vect F = \<P_y-N_z,M_z-P_x,N_x-M_y\>\) describes the counter-clockwise rotation of the field around the \(\veci,\vecj,\veck\) vectors.
    • \((\curl\vect F)\cdot \vect u\) gives the rotation of the field around the unit vector \(\vect u\).
  • The expansion of a vector field is given by divergence.
    • The value \(\dv \vect F=\nabla\cdot\vect F =M_x+N_y+P_z\) sums up how each component of \(\vect F\) grows as each coordinate is increased.

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 15.2, 15.6, 15.7

Exercises

Find the curl and divergence of each of the following vector fields, then find the value of curl and divergence at the given point.

  1. \(\vect F=\<x,y,z\>\) at the origin.
  2. \(\vect F=\<z^2-y^2,x^2-z^2,y^2-x^2\>\) at \(\<1,1,1\>\)
  3. \(\vect F=\<z-x^2,ye^z,xy\>\) at \(\<2,-1,0\>\)
  4. \(\vect F=\<-y^3,2x^3\>\) at \(\<1,1\>\)

Solutions

  1. \(\curl \vect F=\vect 0\), so \(\curl \vect F=\vect 0\) at the origin. \(\dv \vect F=3\), so \(\dv \vect F=3\) at the origin.
  2. \(\curl \vect F=\<2y+2z,2z+2x,2x+2y\>\), so \(\curl \vect F=\<4,4,4\>\) at \(\<1,1,1\>\). \(\dv \vect F=0\), so \(\dv \vect F=0\) at \(\<1,1,1\>\).
  3. \(\curl \vect F=\<x-ye^z,1-y,0\>\), so \(\curl \vect F=\<3,2,0\>\) at \(\<2,-1,0\>\). \(\dv \vect F=-2x+e^z\), so \(\dv \vect F=-3\) at \(\<2,-1,0\>\).
  4. \(\curl \vect F=(6x^2+3y^2)\veck\), so \(\curl \vect F= 9 \veck\) at \(\<1,1\>\). \(\dv \vect F= 0\), so \(\dv \vect F=0\) at \(\<1,1\>\).