{"id":468,"date":"2023-08-27T16:32:59","date_gmt":"2023-08-27T11:02:59","guid":{"rendered":"https:\/\/moodle.sit.ac.in\/blog\/?p=468"},"modified":"2024-05-14T05:25:11","modified_gmt":"2024-05-13T23:55:11","slug":"mathematics-i-for-cse-stream-lab-component","status":"publish","type":"post","link":"https:\/\/moodle.sit.ac.in\/blog\/mathematics-i-for-cse-stream-lab-component\/","title":{"rendered":"Mathematics-I for CSE stream Lab Component"},"content":{"rendered":"\n

(Course Code: BMATS101)<\/mark> solutions using Python Programming<\/p>\n\n\n\n

<\/p>\n\n\n\n

Hi everyone! This blog provides solutions for the lab component of the newly introduced subject “Mathematics-I for Computer Science and Engineering stream<\/strong>” (BMATS101) for I semester students of VTU. I have provided solutions for the lab exercises using Python. All the programs shown below have been tested on Python 3.11.7<\/strong> on Ubuntu 22.04<\/strong> OS using Anaconda Distribution<\/strong>. <\/p>\n\n\n\n

Installation Instructions<\/h2>\n\n\n\n

A detailed procedure of installing ANACONDA <\/strong>and necessary tools for this is provided in my other blog listed below. https:\/\/moodle.sit.ac.in\/blog\/setting-up-anaconda-python-programming-environment\/<\/a><\/p>\n\n\n\n

Questions<\/h2>\n\n\n\n
    \n
  1. 2D plots for Cartesian and polar curves<\/a><\/li>\n\n\n\n
  2. Finding angle between polar curves, curvature and radius of curvature of a given curve<\/a><\/li>\n\n\n\n
  3. Finding partial derivatives and Jacobian<\/a><\/li>\n\n\n\n
  4. Applications to Maxima and Minima of two variables<\/li>\n\n\n\n
  5. Solution of first-order ordinary differential equation and plotting the solution curves<\/a><\/li>\n\n\n\n
  6. Finding GCD using Euclid\u2019s Algorithm<\/a><\/li>\n\n\n\n
  7. Solving linear congruences \ud835\udc82\ud835\udc99 \u2261 \ud835\udc83(\ud835\udc8e\ud835\udc90\ud835\udc85 \ud835\udc8e)<\/a><\/li>\n\n\n\n
  8. Numerical solution of system of linear equations, test for consistency and graphical representation<\/a><\/li>\n\n\n\n
  9. Solution of system of linear equations using Gauss-Seidel iteration<\/a><\/li>\n\n\n\n
  10. Compute Eigen values and Eigen vectors and find the largest and smallest Eigen value by Rayleigh power method.<\/a><\/li>\n<\/ol>\n\n\n\n
    \n\n\n\n
    \"\"<\/figure>\n\n\n\n

    2D plots for Cartesian and Polar curves<\/h2>\n\n\n\n

    The following Python programs prompt the user to choose between Cartesian and Polar curves. It then generates those curves and plots the curve. We have taken two curves one in Cartesian and another in Polar form.<\/p>\n\n\n\n

    The Cartesian curve chosen for this example is $ y = \\sin(2*x) * \\cos(3*x) $<\/strong><\/p>\n\n\n\n

    The Polar curve chosen for this example is $ r = a\\cos(n\\theta) $ with a = 1 and n = 6<\/p>\n\n\n\n

    Python Code<\/h3>\n\n\n\n
    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    import<\/span> <\/span>matplotlib<\/span>.<\/span>pyplot<\/span> <\/span>as<\/span> <\/span>plt<\/span><\/span>\nimport<\/span> <\/span>numpy<\/span> <\/span>as<\/span> <\/span>np<\/span><\/span>\n<\/span>\noption<\/span> = <\/span>int<\/span>(<\/span>input<\/span>(<\/span>"<\/span>1.Cartesian Curve<\/span>\\n<\/span>2.Polar Curve<\/span>\\n<\/span>Choose the option : <\/span>"<\/span>))<\/span><\/span>\n<\/span>\nif<\/span> <\/span>option<\/span> == 1:<\/span><\/span>\n    # <\/span>Define<\/span> <\/span>the<\/span> <\/span>x<\/span> <\/span>values<\/span><\/span>\n    <\/span>x<\/span> = <\/span>np<\/span>.<\/span>linspace<\/span>(-5<\/span>,<\/span> 5<\/span>,<\/span> 100)<\/span><\/span>\n    <\/span><\/span>\n    # <\/span>Define<\/span> <\/span>the<\/span> <\/span>function<\/span> <\/span>to<\/span> <\/span>plot<\/span> (<\/span>y<\/span> = <\/span>sin<\/span>(2<\/span>x<\/span>) + <\/span>cos<\/span>(3<\/span>x<\/span>))<\/span><\/span>\n    <\/span>y<\/span> = <\/span>np<\/span>.<\/span>sin<\/span>(2<\/span>*<\/span>x<\/span>) <\/span>*<\/span> <\/span>np<\/span>.<\/span>cos<\/span>(3<\/span>*<\/span>x<\/span>)<\/span><\/span>\n    #<\/span>another<\/span> <\/span>example<\/span><\/span>\n    # <\/span>y<\/span> = <\/span>x<\/span>**<\/span>3 + <\/span>x<\/span>**<\/span>2<\/span><\/span>\n    # <\/span>Create<\/span> <\/span>the<\/span> <\/span>plot<\/span><\/span>\n    <\/span>plt<\/span>.<\/span>plot<\/span>(<\/span>x<\/span>,<\/span> <\/span>y<\/span>)<\/span><\/span>\n    <\/span><\/span>\n    # <\/span>Add<\/span> <\/span>grid<\/span><\/span>\n    <\/span>plt<\/span>.<\/span>grid<\/span>(<\/span>True<\/span>)<\/span><\/span>\n    <\/span><\/span>\n    # <\/span>Add<\/span> <\/span>labels<\/span> <\/span>and<\/span> <\/span>title<\/span><\/span>\n    <\/span>plt<\/span>.<\/span>xlabel<\/span>(<\/span>'<\/span>X Axis<\/span>'<\/span>)<\/span><\/span>\n    <\/span>plt<\/span>.<\/span>ylabel<\/span>(<\/span>'<\/span>Y Axis<\/span>'<\/span>)<\/span><\/span>\n    <\/span>plt<\/span>.<\/span>title<\/span>(<\/span>'<\/span>Cartesian Curve<\/span>'<\/span>)<\/span><\/span>\n    <\/span><\/span>\n    # <\/span>Show<\/span> <\/span>the<\/span> <\/span>plot<\/span><\/span>\n    <\/span>plt<\/span>.<\/span>show<\/span>()<\/span><\/span>\n    <\/span><\/span>\nelif<\/span> <\/span>option<\/span> == 2:<\/span><\/span>\n    # <\/span>Define<\/span> <\/span>the<\/span> <\/span>theta<\/span> <\/span>values<\/span> <\/span>for<\/span> <\/span>the<\/span> <\/span>rose<\/span> <\/span>curve<\/span><\/span>\n    <\/span>theta<\/span> = <\/span>np<\/span>.<\/span>linspace<\/span>(0<\/span>,<\/span> 2<\/span>*<\/span>np<\/span>.<\/span>pi<\/span>,<\/span> 1000)<\/span><\/span>\n    <\/span><\/span>\n    # <\/span>Define<\/span> <\/span>the<\/span> <\/span>parameters<\/span> <\/span>for<\/span> <\/span>the<\/span> <\/span>rose<\/span> <\/span>curve<\/span><\/span>\n    <\/span>a<\/span> = 1<\/span><\/span>\n    <\/span>n<\/span> = 6<\/span><\/span>\n    <\/span><\/span>\n    # <\/span>Calculate<\/span> <\/span>the<\/span> <\/span>radius<\/span> <\/span>values<\/span> <\/span>for<\/span> <\/span>the<\/span> <\/span>rose<\/span> <\/span>curve<\/span><\/span>\n    <\/span>r<\/span> = <\/span>a<\/span>*<\/span>np<\/span>.<\/span>cos<\/span>(<\/span>n<\/span>*<\/span>theta<\/span>)<\/span><\/span>\n    <\/span><\/span>\n    # <\/span>Create<\/span> <\/span>a<\/span> <\/span>polar<\/span> <\/span>plot<\/span> <\/span>of<\/span> <\/span>the<\/span> <\/span>rose<\/span> <\/span>curve<\/span><\/span>\n    <\/span>plt<\/span>.<\/span>polar<\/span>(<\/span>theta<\/span>,<\/span> <\/span>r<\/span>)<\/span><\/span>\n    <\/span>plt<\/span>.<\/span>title<\/span>(<\/span>"<\/span>Polar Curve<\/span>"<\/span>)<\/span><\/span>\n    # <\/span>Show<\/span> <\/span>the<\/span> <\/span>plot<\/span><\/span>\n    <\/span>plt<\/span>.<\/span>show<\/span>()<\/span><\/span>\n<\/span><\/code><\/pre><\/div>\n\n\n\n
    Output Sample 1<\/h3>\n\n\n\n
    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    1<\/span>.<\/span>Cartesian<\/span> <\/span>Curve<\/span><\/span>\n2<\/span>.<\/span>Polar<\/span> <\/span>Curve<\/span><\/span>\nChoose<\/span> <\/span>the<\/span> option <\/span>:<\/span> <\/span>1<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n
    \"1.Cartesian<\/a><\/figure>\n\n\n\n
    Output Sample 2<\/h3>\n\n\n\n
    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    1<\/span>.<\/span>Cartesian<\/span> <\/span>Curve<\/span><\/span>\n2<\/span>.<\/span>Polar<\/span> <\/span>Curve<\/span><\/span>\nChoose<\/span> <\/span>the<\/span> option <\/span>:<\/span> <\/span>2<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n
    \"\"<\/a><\/figure>\n\n\n\n
    \n\n\n\n
    Finding angle between polar curves, curvature and radius of curvature of a given curve<\/h2>\n\n\n\n
    Finding angle between polar curves<\/h3>\n\n\n\n
    Here we show how to find the angle between two polar curves, for this we consider the following two polar curves<\/p>\n\n\n\n