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flyinjoe

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Thank you

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In summary, the conversation discusses the calculation of the basis of a surface's tangent vector. The expert provides a method for finding a basis by using the normal vector and two independent vectors in the tangent plane. The asker is grateful for the concise and helpful explanation.

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flyinjoe

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Thank you

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HallsofIvy

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For example, if the surface is given by [itex]x^2yz= 1[/itex] then the normal vector at any point is [itex]<2xyz, x^2z, x^2y>. At (1, 1, 1) that would be <2, 1, 1>. The tangent plane there is 2(x- 1)+ y- 1+ z- 1= 0 or 2x+ y+ z= 4. if x= 2, y= 1, then 4+ 1+ z= 1 so z= -1. The point (2, 1, -1) is also in that plane so the vector <2- 1, 1- 1, 1-(-1)>= <1, 0, 2> lies in that tangent plane. If x= 1, y= 2, then 2(0)+ 1+ z= 4 so z= 3. The point (1, 2, 3) is also in that tangent plane so the vector <1- 1, 2- 1, 3-(-1)>= <0, 1, 4> lies in that tangent plane. The two vectors <1, 0, 2> and <0, 1, 4> are two independent vectors in thet tangent plane and so form a basis.

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flyinjoe

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Thank you!

The basis of a tangent plane is the set of two linearly independent vectors that span the plane. These vectors are used to describe the orientation and shape of the plane at a specific point on a curved surface.

The basis of a tangent plane can be calculated by taking the partial derivatives of the surface equation at a specific point and using them as the coefficients for the basis vectors. These vectors can then be normalized to ensure they are linearly independent and perpendicular to each other.

Calculating the basis of a tangent plane is important because it allows us to approximate a curved surface with a flat plane at a specific point. This approximation is useful in various fields such as physics, engineering, and computer graphics.

The basis of a tangent plane is always perpendicular to the normal vector of the surface at a specific point. This means that the basis vectors can be used to determine the direction of the surface's curvature, while the normal vector describes the direction perpendicular to the surface.

Yes, the basis of a tangent plane can change at different points on a surface. This is because the orientation and shape of a surface can vary from point to point, and the basis vectors are dependent on these factors. Therefore, the basis of a tangent plane must be recalculated for each point on the surface.

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