Example: Tangent Planes and Gradient Descent

with(plots): with(geom3d): f := 4*x^2 - 5*x*y + 3*y^2 ;

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plane1 := plot3d(f,x=-3..3,y=-3..3,axes=normal): display( { plane1 }) ;

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tangentEq := 3*x + y - 2 : T := plot3d(tangentEq ,x=-3..3,y=-3..3): display({ T, plane1 });

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restart ; with(plots): with(plottools) : with(LinearAlgebra): f := 4*x^2 - 5*x*y + 3*y^2 ;

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fx := diff(f,x); fy := diff(f,y);

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gradient_f := <fx,fy>;

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x0 := -1; y0 := 1;

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

g0 := Normalize(subs({x=x0,y=y0},gradient_f));

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contour_plot := contourplot(f,x=-3..3,y=-3..3): gradient_line := line([x0,y0],[x0+g0[1],y0+g0[2]]): display({contour_plot,gradient_line});

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restart; with(plots): with(plottools):

Warning, the name changecoords has been redefined

Warning, the assigned name arrow now has a global binding

f := - 4*(x-2)^2 + x*y - 4*y^2 ;

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smartplot3d[x,y](f);

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contourplot(f,x=-1..5,y=-3..3,contours=20,filled=true);

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VDwiRi43JCQiM2lxKClHTGl3JD4jRi4kITNycCgpR0xpd2A2Ri43JDckRmp3JCEzKFFBbiwhKW8/NiJGLkZdYG03JEZmX203JCQiM0swaSZIIik+IT5BRi4kIjNzJnpWcT0hKTRxIkYuNyQ3JEZqdyQiM2d1QSJlS2AlKnAiRi5GZ2BtNyRGY2BtNyQkIjMiMzliUT8uQ1EjRi4kITMhKlJeJlE/LkM1IkYuNyQ3JCQiMy9TNiNRJ2UmelYjRi5GZWlsRmFhbTckRmdhbTckJCIzUmYkZig0dXFZREYuJCEzInlOZig0dXFFNUYuNyQ3JEZqeSQhM3AjPUhEIWVXLTVGLkZbYm03JDckRmp3JCIzT3VBImVLYCUqcCJGLjckJCIzYCM0NE1TKGYnWyNGLiQiMyYqMzRmJ2YtTW4iRi43JDckRmp5JCIzNiYpUl9bNjFfO0YuRmhibTckRmFibTckJCIzKSlHNXghUlZ2cCNGLiQhM152LXIyUlZ2JCpGMTckNyRGaltsJCEzOiVmTVZzJXAiWylGMUZiY203JDckRmpbbCQiMz8pKmU7VTglKWU6Ri43JCQiM3opPXcvVzB3JEdGLiQiM3MrKysrKytnOkYuNyRGX2RtRl5jbTckRmhjbTckJCIzJ2VDLG5oZFclR0YuJCEzJjNXN3E7d1hXKUYxNyQ3JCQiMydvOG4qKT0oM11HRi5GZ2dsRmZkbTckNyQkIjNKUHInKik9KDNdR0YuRmdnbDckJCIzNSpmW1J4IWVsSEYuJCEzQ3RmW1J4IWVEKEYxNyQ3JCQiMyc0KysrKysrMyRGLiQhM1hEamdZIyk0JT0nRjFGY2VtNyQ3JEZqW2wkIjMpeiplO1U4JSllOkYuNyQkIjMpNEdbJXlcTFdHRi4kIjNFPzxiQF1tYjpGLjckNyRGamVtJCIzJFwhM2BlKnpaUSJGLkZiZm03JEZpZW03JCQiMyo+KDM+PjVXKTMkRi4kITNlLygzPj41VzMnRjE3JDckJCIzKSlcRFAiPWJmNCRGLkZlcEZcZ203JEZiZ203JCQiM1ReeVFgYF4nPSRGLiQhM1QlXHlRYGBebSVGMTckNyQkIjNvL2hfSm0zaktGLkY1RmZnbTckRlxobTckJCIzW3YheSN6JTNARyRGLiQhM0JPMnkjeiUzQEtGMTckNyQkIjM8LCsrKysrP0xGLiQhM2suOiFlYTYjPURGMUZgaG03JDckRmdobSQiMz8nb2NCJXlpITQiRi43JCQiMyE9M3BedUpnOSRGLkZlamw3JEZfaW1GaGZtNyQ3JEZnaG0kITMzLjohZWE2Iz1ERjE3JCQiMyZleSkzSHJNZ0xGLiQhM05QeSkzSHJNZyJGMTckNyQkIjM2KClSQ3BATXpMRi5GQUZnaW03JDckJCIzbicpUkNwQE16TEYuRkE3JCQiM15KbVsjb00lPk1GLiQiM0NecUxePGBjP0ZJNyQ3JCQiM2UhRzdcQCdbXU1GLiQiM2UyKysrKysrN0YxRmRqbTckNyRGW1tuRk43JCQiM1FdaTomR2tDWSRGLiQiMyN5XlAlW3JOdkBGMTckNyQkIjNDME0mcCopKlIkWyRGLkZaRmFbbjckRmdbbjckJCIzUk9aKkdHYyd6TUYuJCIza19FMHJyVi5XRjE3JDckJCIzPiNHXj9gaWJaJEYuRmBvRltcbjckRmFcbjckJCIzdm4pPWIlb0VeTUYuJCIzM1k4IlthSnQzKEYxNyQ3JCQiMzpLVWhlb0FDTUYuRmJxRmVcbjckRltdbjckJCIzWlc7YHgwY0lMRi4kIjNtZCRvQ1VSJXA1Ri43JDckJCIzIyo9XUgkKkhSRUxGLkZkc0ZfXW43JEZlXW5GXGltLUYkNmJxNyQ3JCQiM1kqKioqKioqKioqKioqUiVGMSQhMyJ5bjQoUSlbI1F0RjE3JCQiM0E7Oyl5MjBRdiRGMUZlcDckNyQkIjNtOjspeTIwUXYkRjFGZXA3JCQiMyIpZWpqakc5ZE1GMSQhM05eampqRzlkXUYxNyQ3JCQiM2giKmZfazJNISlIRjFGNUZoXm43JEZeX243JCQiM2IqUiYpZjlOZHEjRjEkITMhPlImKWY5TmQhPkYxNyQ3JCQiM2kkKVwpXClccCFlI0YxRkFGYl9uNyQ3JCQiMzEkKVwpXClccCFlI0YxRkE3JCQiMyp5RUhvbmhIYiNGMSQiM3InUjI8QiRRcWtGSTckNyQkIjM6QTtibio9XWAjRjFGTkZfYG43JEZlYG43JCQiMyJcYk11UV5cdSNGMSQiM0dfYWM3J1tdJkdGMTckNyQkIjN4Im9QWXNXWyNHRjFGWkZpYG43JEZfYW43JCQiMyFwcVpzPi9POiRGMSQiM2wqSF9GIWVSWVtGMTckNyQkIjMpei5QcS5uSFYkRjFGYG9GY2FuNyRGaWFuNyQkIjNUaUlsSycpRzJQRjEkIjMvV3BNbjhyI3AnRjE3JDckJCIzJHlkTipRdExWVkYxRmJxRl1ibjckNyQkIjNGeGIkKlF0TFZWRjFGYnE3JCQiM3Fza25HXiJHTyVGMSQiM0JNTktyWz1QJSlGMTckNyRGXV5uJCIzWD4nW18wZzxeKUYxRmpibjckNyRGYGFsJCEzaDcnMylmMCd5MSJGLjckJCIzJSpSUCsoUnNqLCZGMUZnZ2w3JDckJCIzMFRQKyhSc2osJkYxJCEzLyQqKioqKioqKioqKipSKUYxRlxebjckNyQkIjMjKikqKioqKioqKioqKipSJUYxJCIzTT0nW18wZzxeKUYxNyQkIjNlS2xLO3lDWF9GMSQiM0R1TW4kPV9aJioqRjE3JDckJCIzT1R6XU8qSFtyJkYxRmRzRmZkbjckRlxlbjckJCIza1AjKTNlJ1E7PCdGMSQiMyFvPCI+TWgkRzkiRi43JDckRmBhbCQiM2YqM2dVTidHTDdGLkZgZW43JDckRmFlbCQhM1k0XywjcERaRSJGLjckJCIzd3FjJFtwOSI+cEYxRmVpbDckRl1mbjckJCIzRykqKioqKioqKioqKip6J0YxRmVjbjckRmZlbjckJCIzb3QmRzlkTlk+KEYxJCIzM1ZyJkdXTzBHIkYuNyQ3JCQiM05oLVRjLUszdkYxRmVqbEZlZm43JEZbZ243JCQiMyE9NyV6QS8wXiQpRjEkIjNxKWU/eCZcKltTIkYuNyQ3JEZhZWwkIjNlcmEyS1xGJVwiRi5GX2duNyQ3JEYsJCEzP2d4aWArXyVSIkYuNyQkIjM/VXFLcyZHOSwiRi4kITM8KioqKioqKioqKioqPjhGLjckRlxobkZqZW43JDckRmFlbCQiMyE9WnY/JFxGJVwiRi43JCQiM0UnPWZ6Kj1YYiYqRjEkIjMtIzMvLSJbWEM6Ri43JDckJCIzTyQpKWVXKCkpKVImKipGMUZiZG1GZmhuNyRGXGluNyQkIjMzNyV6QS8wLDQiRi4kIjNOKWU/eCZcKilIO0YuNyQ3JEYsJCIzIVJSKFJhSCZRbyJGLkZgaW43JEZpZ243JCQiM1E9WnZxJkc5RiJGLiQhM2o8WnZxJkc5ViJGLjckNyRGXnAkITM3PDpQZ0RodjlGLkZqaW43JDckRiwkIjM3JVIoUmFIJlFvIkYuNyQkIjMuUD1meidwRkIiRi4kIjNnaiIzL0tJc3MiRi43JDckJCIzS0xeRktAbVs4Ri4kIjNzKysrKysrKz1GLkZnam43JEZdW283JCQiMytDKTNlJ1E7I1EiRi4kIjNpdzY+TWgkeSI9Ri43JDckRl5wJCIzNyFcQEQpPSF5Iz1GLkZjW283JEZgam43JCQiM0lkRC00ImUqKWUiRi4kITN5Y0QtNCJlKjM6Ri43JDckJCIzOCsrKysrK1M7Ri4kITNiJyplOSkzSyg9OkYuRl1cbzckNyQkIjM3KysrKysrKzlGLkZqW283JCQiM2U3MWBFcTheOkYuJCIzIXlRcE0oSCcpKSk9Ri43JDckRmJ0JCIzSm9pJlw1WnAjPkYuRlxdbzckNyRGZnIkITN4JyplOSkzSyg9OkYuNyQkIjMpMytEIkc5S1c9Ri4kITM7K103RzlLQzpGLjckNyRGW3UkITMoSFM+aCgqPWZfIkYuRmldbzckNyRGZnJGY11vNyQkIjNIWmhQLCVRKkc8Ri4kIjNKYFFpKWZoNSY+Ri43JDckRlt1JCIzZUByIVt1UWEqPkYuRmRebzckRl9ebzckJCIzKTRCNjw5WVcxI0YuJCEzIilIN3JUaFcvOkYuNyQ3JEZkdiQhMyVHJG8oPScqcCEqXCJGLkZeX283JDckRlt1JCIzIT03Mlt1UWEqPkYuNyQkIjN5ZCxuIyo9Sj4+Ri4kIjMvVilIdDUpbys/Ri43JDckRmR2JCIzQWEieilSKTQ7LiNGLkZbYG83JEZkX283JCQiM3c5YmpFLW5pQUYuJCEzJFFeTm1BcUVZIkYuNyQ3JEZqdyQhM3EneVRsZl0qUjlGLkZlYG83JEZhYG83JCQiMyNcQyFSKipvWUdARi4kIjM3Yyg0MTVMOi4jRi43JDckRmp3JCIzPXAyQkgubkw/Ri5GX2FvNyRGW2FvNyQkIjNRTlA1dldOWUNGLiQhM0NNUDV2V04xOUYuNyQ3JEZqeSQhMzcweHckUjEtTiJGLkZpYW83JEZlYW83JCQiMzslKXk7QHldbkJGLiQiM2s7QCQpeUBcSz9GLjckNyRGankkIjNDRyEqW3dhcCoqPkYuRmNibzckRl9ibzckJCIzLXVgIyoqUSQqKT5FRi4kITMpR1BEKipRJCopUjhGLjckNyQkIjNpL04lKkhAQ2dFRi5GX2huRl1jbzckNyQkIjM9L04lKkhAQ2dFRi5GX2huNyQkIjMwP2IoKUgkKilSeCNGLiQhM0U9YigpSCQqKVJEIkYuNyQ3JEZqW2wkITNfJClcJl