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Chapter 11: Problem 32

Give the standard form of the equation of the tangent plane to a surface given by \(F(x, y, z)=0\) at \(\left(x_{0}, y_{0}, z_{0}\right)\).

Short Answer

Expert verified
The standard form of the equation of the tangent plane to a surface given by \(F(x, y, z)=0\) at \((x_0,y_0, z_0)\) is \(\frac{\partial F}{\partial x}(x_0, y_0,z_0)(x - x_0) + \frac{\partial F}{\partial y}(x_0, y_0,z_0)(y - y_0) + \frac{\partial F}{\partial z}(x_0, y_0,z_0)(z - z_0) = 0\).

Step by step solution

01

Compute the Gradient

Find the gradient (鈭嘑) of the function \(F(x, y, z)\). The gradient is given by \(\left(\frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z}\right)\).
02

Evaluate the Gradient at the given point

Evaluate gradient of \(F(x, y, z)\) at the given point \((x_{0}, y_{0}, z_{0})\). Let's call this vector \(N\). Thus, \(N = (鈭嘑)(x_0, y_0,z_0) = \left(\frac{\partial F}{\partial x}(x_0, y_0,z_0), \frac{\partial F}{\partial y}(x_0, y_0,z_0), \frac{\partial F}{\partial z}(x_0, y_0,z_0)\right)\).
03

Formulate the Equation of the Tangent Plane

The equation of the tangent plane at \((x_0,y_0, z_0)\) can be written using the dot product of the normal vector with the vector laying on the plane. It is given by \(N \cdot (x - x_0, y - y_0, z - z_0) = 0\). Taking the dot product and arranging the equation gives the standard form of the equation of the tangent plane: \(\frac{\partial F}{\partial x}(x_0, y_0,z_0)(x - x_0) + \frac{\partial F}{\partial y}(x_0, y_0,z_0)(y - y_0) + \frac{\partial F}{\partial z}(x_0, y_0,z_0)(z - z_0) = 0\). This equation represents a tangent plane to the surface specified by the function \(F(x, y, z)\) at the point \((x_0,y_0, z_0)\).

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