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.

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

### Solutions

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