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Mathematics LibreTexts

13.3: Visualizing Two-Dimensional Scalar and Vector Field

  • Page ID
    7845
  • [ "article:topic", "Vector Fields", "authorname:hsayama", "scalar field" ]

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    Plotting scalar and vector fields in Python is straightforward, as long as the space is two-dimensional.  Here is an example of how to plot a 3-D surface plot:

    code 13.1.png

    The scalar field \(f(x,y) = \sin{\sqrt{x^2 + y^2}}\) is given on the right hand side of the zvalues part. The result is shown in Fig. 13.3.1.

    fig 13.6.png

    Figure \(\PageIndex{1}\):  Scalar field visualized as a 3-D surface using Code 13.1.

    And here is how to draw a contour plot of the same scalar field:

    code 13.2.png
    The clabel command is used here to add labels to the contours. The result is shown in Fig. 13.3.2.

    fig 13.7.png

    Figure \(\PageIndex{2}\): Scalar field visualized as a contour plot using Code 13.2

    If you want more color, you can use imshow, which we already used for CA:

    code 13.3 1.png

    code 13.3 2.png
    The result is shown in Fig. 13.3.3. Colorful!

    fig 13.8.png

    Figure \(\PageIndex{3}\): Scalar field visualized as a color image using Code 13.3.

    Finally, a two-dimensional vector field can be visualized using the streamplot function that we used in Section 7.2. Here is an example of the visualization of a vector field v = (vx,vy) = (2x,y−x), with the result shown in Fig. 13.3.4:

    code 13.4.png

    code 13.4 2.png

    fig 13.9.png

    Figure \(\PageIndex{4}\): Vector field visualized using Code 13.4.

    Exercise  \(\PageIndex{1}\): 

    Plot the scalar field \(f(x,y) = \sin{(xy)}\) for \(−4 ≤ x,y ≤ 4\) using Python.

    Exercise  \(\PageIndex{2}\): 

    Plot the gradient field of f\((x,y) = \sin{(xy)}\) for \(−4 ≤ x,y ≤ 4\) using Python.

    Exercise  \(\PageIndex{3}\): 

    Plot the Laplacian of \(f(x,y) = \sin{(xy)}\) for \(−4 ≤ x,y ≤ 4\) using Python. Compare the result with the outputs of the exercises above.