Level Curves

Caution: Caution: TEMATH's tools will write the values of computed results at the bottom of this Report window. This will cause you to constantly scroll between the computed results and the instructions for this activity. You may want to print a copy of the contents of the Report window before you begin this activity.

Example 1: z = f(x, y) = sin(x) + sin(y)

Select New Function from the Work menu.
"y1(x) = " is written into the top cell of the Work window.
Change the argument list of the default function y1(x) from one variable to two variables y1(x, y) by inserting ",y" after the x to get "y1(x,y) = ".
Type the expression "sin(x) + sin(y)" for the function after the equal sign: y1(x,y) = sin(x) + sin(y).
Press the Enter key
Select Plot from the Graph menu.

The black curves represent positive values of z and the red curves represent negative values of z. As you move the cursor around the Graph window, note that the values of x, y, and z are given in the bottom three cells of the Domain & Range window. Additionally, an estimate of the min and max for the z values over the plot rectangle is written into the report window.

To add a level curve for a particular value of z,

Move the cursor around the Graph window until you find the value of z that you want.
Click the mouse button and the level curve will be drawn. You can click as many level curves as you want.

You can change the default number of level curves drawn, the range of z values, and the thickness of the pen used for drawing the added level curves by selecting Level Curves... from the Options menu. Also, you can increase the grid size for more accurate plots but this will slow down the plotting speed.

To plot color and pattern density plots,

Select Level Curves... from the Options menu. Click the Color Density button. Click the OK button.
Select Plot from the Graph menu.
Select Level Curves... from the Options menu. Click the Pattern Density button. Click the OK button.
Select Plot from the Graph menu.

Example 2: z = f(x, y) = -3y/(x^2+y^2+1)

Select Level Curves... from the Options menu. Click the Level Curves button. Click the OK button.
Select New Function from the Work menu.
Enter the following function into the Work window: y2(x,y) = -3y/(x^2+y^2+1)
Select Plot from the Graph menu.

Example 3: Level Curves with an Artistic Flavor

Select New Function from the Work menu.
Enter the following function into the Work window: y3(x,y) = sin(x)^3 + cos(y)^2
Select Plot from the Graph menu.
Select New Function from the Work menu.
Enter the following function into the Work window: y4(x,y) = sin(2x+y)^3 + cos(2y-x)^2
Select Plot from the Graph menu.

Example 4: Equipotential Curves

The electric potential at a point (x, y) is given by V(x, y) = 1/sqrt(25 – x^2 - y^2). Plot some equipotential curves.

Select New Function from the Work menu.
Enter the following function into the Work window: y5(x,y) = 1/sqrt(25 - x^2 - y^2)
Select Plot from the Graph menu.

Example 5: Isothermals

The temperature of a metal plate at the point (x, y) is given by T(x, y) = 50/(1 + 3x^2 + y^2). Plot some isothermals.

Select New Function from the Work menu.
Enter the following function into the Work window: y6(x,y) = 50/(1 + 3x^2 + y^2)
Select Plot from the Graph menu.

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