# MA 227 Standard C10

Vector Fields

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 $$\<1,1\>$$.