How would you plot the gradient vector field of f(x,y)=x^2-3*x*y+y-y^2 on the domain [-4,4]*[-4,4] in Mathematica 7?
I know the function is "VectorPlot" (h t t p : / / g o o . g l / p c S f), but I don't know how to isolate the x and y inputs because of the -3*x*y term in the function.
To plot the gradient vector field of the function f(x, y) = x^2 - 3xy + y - y^2 on the domain [-4,4]x[-4,4] in Mathematica 7, you can use the VectorPlot function.
The first step is to compute the gradient of the function f(x, y). The gradient is a vector that points in the direction of the greatest rate of increase of the function. In Mathematica, you can do this using the Grad function, which takes the function f(x, y) as an argument. The gradient vector is given by:
gradient = Grad[f[x, y], {x, y}]
Next, you need to define the domain of the plot. In this case, it is the rectangle [-4,4]x[-4,4]. You can do this using the RegionFunction option of VectorPlot:
domain = {{-4, 4}, {-4, 4}}
Now, you are ready to plot the gradient vector field. Use the VectorPlot function, specifying the gradient vector as the first argument, and the domain as the second argument:
VectorPlot[gradient, domain]
This will plot the gradient vector field of the function f(x, y) = x^2 - 3xy + y - y^2 on the domain [-4,4]x[-4,4].