Tangent Plane - Revision history

Lila: /* Licensing */

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Licensing

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Resources

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Tangent Plane

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 Content obtained and/or adapted from:
 * [https://openstax.org/books/calculus-volume-3/pages/4-4-tangent-planes-and-linear-approximations Tangent planes and linear approximations, OpenStax] under a CC BY-SA license
-* [https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Vector_Calculus_(Corral)/02%3A_Functions_of_Several_Variables/2.03%3A_Tangent_Plane_to_a_Surface Tangent Plane to a Surface], Mathematics LibreTexts] under a CC BY-SA license
+* [https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Vector_Calculus_(Corral)/02%3A_Functions_of_Several_Variables/2.03%3A_Tangent_Plane_to_a_Surface Tangent Plane to a Surface, Mathematics LibreTexts] under a CC BY-SA license

← Older revision		Revision as of 21:33, 2 November 2021
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	*[https://www.youtube.com/watch?v=e1Kp-fUIJCU Tangent Plane Approximations] Video by -patrickJMT		*[https://www.youtube.com/watch?v=e1Kp-fUIJCU Tangent Plane Approximations] Video by -patrickJMT
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://openstax.org/books/calculus-volume-3/pages/4-4-tangent-planes-and-linear-approximations Tangent planes and linear approximations, OpenStax] under a CC BY-SA license
		+	* [https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Vector_Calculus_(Corral)/02%3A_Functions_of_Several_Variables/2.03%3A_Tangent_Plane_to_a_Surface Tangent Plane to a Surface], Mathematics LibreTexts] under a CC BY-SA license

← Older revision		Revision as of 17:28, 6 October 2021
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	'''Definition of Tangent Plane'''		'''Definition of Tangent Plane'''
		+	[[File:Tangent plane.jpg\|thumb\|Tangent plane]]
	: Let <math> P_0 = (x_0,y_0,z_0) </math> be a point on a surface <math>S</math>, and let <math>C</math> be any curve passing through <math>P_0</math> and lying entirely in <math>S</math>. If the tangent lines to all such curves <math>C</math> at <math>P_0</math> lie in the same plane, then this plane is called the tangent plane to <math>S</math> at <math>P_0</math>.		: Let <math> P_0 = (x_0,y_0,z_0) </math> be a point on a surface <math>S</math>, and let <math>C</math> be any curve passing through <math>P_0</math> and lying entirely in <math>S</math>. If the tangent lines to all such curves <math>C</math> at <math>P_0</math> lie in the same plane, then this plane is called the tangent plane to <math>S</math> at <math>P_0</math>.

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 Intuitively, it seems clear that, in a plane, only one line can be tangent to a curve at a point. However, in three-dimensional space, many lines can be tangent to a given point. If these lines lie in the same plane, they determine the tangent plane at that point. A tangent plane at a regular point contains all of the lines tangent to that point. A more intuitive way to think of a tangent plane is to assume the surface is smooth at that point (no corners). Then, a tangent line to the surface at that point in any direction does not have any abrupt changes in slope because the direction changes smoothly.
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 : <math> \implies z = f(x_0,y_0) + f_x(x_0,y_0)(x - x_0) + f_y(x_0,y_0)(y - y_0) </math>
 ==Resources==

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 This vector is perpendicular to both lines and is therefore perpendicular to the tangent plane. We can use this vector as a normal vector to the tangent plane, along with the point <math>P_0=(x_0,y_0,f(x_0,y_0))</math> in the equation for a plane:
-<math> n\cdot ((x-x_0)\mathbf{i}+(y-y_0)\mathbf{j}+(z-f(x_0,y_0))\mathbf{k}) = 0 </math>
+: <math> n\cdot ((x-x_0)\mathbf{i}+(y-y_0)\mathbf{j}+(z-f(x_0,y_0))\mathbf{k}) = 0 </math>
-<math> \implies (f_x(x_0,y_0)\mathbf{i}+f_y(x_0,y_0)\mathbf{j}-\mathbf{k})\cdot ((x-x_0)i+(y-y_0)\mathbf{j}+(z-f(x_0,y_0))\mathbf{k}) = 0 </math>
+: <math> \implies (f_x(x_0,y_0)\mathbf{i}+f_y(x_0,y_0)\mathbf{j}-\mathbf{k})\cdot ((x-x_0)i+(y-y_0)\mathbf{j}+(z-f(x_0,y_0))\mathbf{k}) = 0 </math>
-<math> \implies f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)-(z-f(x_0,y_0)) = 0 </math>
+: <math> \implies f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)-(z-f(x_0,y_0)) = 0 </math>
-<math> \implies z = f(x_0,y_0) + f_x(x_0,y_0)(x - x_0) + f_y(x_0,y_0)(y - y_0) </math>
+: <math> \implies z = f(x_0,y_0) + f_x(x_0,y_0)(x - x_0) + f_y(x_0,y_0)(y - y_0) </math>
 ==Resources==
+* [https://openstax.org/books/calculus-volume-3/pages/4-4-tangent-planes-and-linear-approximations Tangent planes and linear approximations], OpenStax
 ===Videos===

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 <math> n\cdot ((x-x_0)\mathbf{i}+(y-y_0)\mathbf{j}+(z-f(x_0,y_0))\mathbf{k}) = 0 </math>
 <math> (f_x(x_0,y_0)\mathbf{i}+f_y(x_0,y_0)\mathbf{j}-\mathbf{k})\cdot ((x-x_0)i+(y-y_0)\mathbf{j}+(z-f(x_0,y_0))\mathbf{k}) = 0 </math>
 <math> f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)-(z-f(x_0,y_0)) = 0 </math>

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 To see why this formula is correct, let’s first find two tangent lines to the surface S. The equation of the tangent line to the curve that is represented by the intersection of S with the vertical trace given by <math>x = x_0</math> is <math>z=f(x_0,y_0)+f_y(x_0,y_0)(y-y_0)</math>. Similarly, the equation of the tangent line to the curve that is represented by the intersection of <math>S</math> with the vertical trace given by <math>y=y_0</math> is <math>z = f(x_0,y_0) + f_x(x_0,y_0)(x - x_0)</math>. A parallel vector to the first tangent line is <math>a = \mathbf{j} + f_y(x_0,y_0)\mathbf{k}</math>; a parallel vector to the second tangent line is <math> b = \mathbf{i} + f_x(x_0,y_0)\mathbf{k} </math>. We can take the cross product of these two vectors:
-: <math>
+:<math>
   a\times b = (\mathbf{j} + f_y(x_0,y_0)\mathbf{k}) \times (\mathbf{i}+f_x(x_0,y_0)\mathbf{k}) </math>
-:::<math> = f_x(x_0,y_0)\mathbf{i}+f_y(x_0,y_0)\mathbf{j} - \mathbf{k} </math>
+::: <math> = f_x(x_0,y_0)\mathbf{i}+f_y(x_0,y_0)\mathbf{j} - \mathbf{k} </math>
 This vector is perpendicular to both lines and is therefore perpendicular to the tangent plane. We can use this vector as a normal vector to the tangent plane, along with the point <math>P_0=(x_0,y_0,f(x_0,y_0))</math> in the equation for a plane:
 ==Resources==