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 - 3*x*y + y - y^2 on the domain [-4,4]x[-4,4] in Mathematica 7, you can follow these steps:
1. Open Mathematica 7 and create a new notebook.
2. Define the function f(x,y) using the Power (^) and Times (*) operators: f[x_, y_] := x^2 - 3*x*y + y - y^2
3. Use the VectorPlot function to plot the gradient vector field. The syntax for VectorPlot is: VectorPlot[{fx, fy}, {x, xmin, xmax}, {y, ymin, ymax}]
- Extract the gradient components by taking the partial derivatives of f with respect to x and y: fx = D[f[x, y], x]; fy = D[f[x, y], y]
- Specify the domain as {x, -4, 4}, {y, -4, 4} to restrict the plot to the desired range.
- The full code for VectorPlot would be: VectorPlot[{fx, fy}, {x, -4, 4}, {y, -4, 4}]
Here's what the code would look like:
```mathematica
f[x_, y_] := x^2 - 3*x*y + y - y^2
fx = D[f[x, y], x];
fy = D[f[x, y], y];
VectorPlot[{fx, fy}, {x, -4, 4}, {y, -4, 4}]
```
Copy and paste this code into the Mathematica 7 notebook, and then run the notebook to generate the plot of the gradient vector field for the given function on the specified domain.