In my lecture it was not clear enough how to use Equation 4.5 to calculate force. I found what I consider a clear explanation online (Equation 5 on this web page and the example immediately following it).
The way the formula is supposed to work is as follows:
Each component of E is some scalar function of position r. To find the x component of the force on a dipole, start by taking the gradient of the function E_x(r).
Now take the dot product of p (the full p vector!) with the gradient you just found. The result is the x component of the force on the dipole.
Repeat for the y and z components of E to find the y and z components of F.
The example on the we page linked about offers an electric field that is basically E=k(y x_hat + x y_hat). To find the x component of force, take the gradient of the function ky, which yields the vector k y_hat. Now dot this with your dipole vector p; this will give k p_y (y_hat dot y_hat) = k p_y.
Similarly, to find the y component of force, take the gradient of the y component of E. This gives k x_hat. The dot product of this gradient vector with p will therefore be kp_x.
Below is the PowerPoint from this afternoon (with a new Slide 10 on this topic). Note that we skipped Problem 4.7 and Problem 4.14 in class.