Question

Use a computer algebra system to graph several representative vectors in the vector field. $$ \mathbf{F}(x, y)=\frac{1}{8}\left(2 x y \mathbf{i}+y^{2} \mathbf{j}\right) $$

Short answer

To represent the vector field graphically, vectors generated by plugging a range of (x,y) coordinates into given vector field, \[F(x, y) = \frac{1}{8}(2 x y * i + y^{2} * j)\], have to be plotted at their respective (x,y) positions. The specifics of generating the (x, y) pairs and the vectors, and the actual plotting, would depend on the particular computer algebra system being used.

Step-by-step solution

Define the vector field function

Using your computer algebra system (for instance, MATLAB, Python, or Wolfram Alpha), define the vector field function \[ F(x, y) = \frac{1}{8}(2 x y * i + y^{2} * j) \]. You typically use 'i' and 'j' in programming to represent unit vectors in the X and Y directions, respectively.

Generate (x, y) pairs

Decide on a range of x and y values that you want to investigate. For example, you might choose -10 to 10 for both x and y. Then, generate a grid of (x, y) pairs covering this range. Your method of doing this will depend on the computer algebra system you're using. In Python's numpy, for example, you would use the meshgrid function.

Evaluate the vector field at each (x, y) pair

Using your computer algebra system, apply the vector field function F to each (x, y) pair in your grid. This will give you a new grid of vectors.

Plot the vector field

You can now plot your vector field using the (x, y) pairs as positions and the evaluated vectors as directions. In most computer algebra systems, this can be done using a quiver plot. The plot will show you how vectors in the vector field behave across your chosen range of x and y values.