Tangent Plane

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Tangent Plane

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.

Definition of Tangent Plane

Tangent plane
Let be a point on a surface , and let be any curve passing through and lying entirely in . If the tangent lines to all such curves at lie in the same plane, then this plane is called the tangent plane to at .

For a tangent plane to a surface to exist at a point on that surface, it is sufficient for the function that defines the surface to be differentiable at that point, defined later in this section. We define the term tangent plane here and then explore the idea intuitively.

Equation of Tangent Plane

Let be a surface defined by a differentiable function , and let be a point in the domain of . Then, the equation of the tangent plane to at is given by

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 is . Similarly, the equation of the tangent line to the curve that is represented by the intersection of with the vertical trace given by is . A parallel vector to the first tangent line is ; a parallel vector to the second tangent line is . We can take the cross product of these two vectors:

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 in the equation for a plane:

Linear Approximations

When working with a function of two variables, the tangent line is replaced by a tangent plane, but the approximation idea is much the same.

Definition of Linear Approximation Given a function with continuous partial derivatives that exist at the point , the linear approximation of at the point is given by the equation

Notice that this equation also represents the tangent plane to the surface defined by at the point . The idea behind using a linear approximation is that, if there is a point at which the precise value of is known, then for values of reasonably close to , the linear approximation (i.e., tangent plane) yields a value that is also reasonably close to the exact value of . Furthermore the plane that is used to find the linear approximation is also the tangent plane to the surface at the point .

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