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Chapter 13: Problem 3

Write an equation for the plane tangent to the surface \(F(x, y, z)=0\) at the point \((a, b, c)\)

Short Answer

Expert verified
Answer: To find the equation of the tangent plane to the surface \(F(x, y, z) = 0\) at a point \((a, b, c)\), follow these steps: 1. Compute the gradient of \(F(x, y, z)\), which is the vector formed by its partial derivatives with respect to \(x\), \(y\), and \(z\). 2. Evaluate the gradient vector at the given point (\(a, b, c\)), which gives us the normal vector of the tangent plane. 3. Write the point-normal form of the tangent plane equation using the evaluated gradient and the given point (\(a, b, c\)). 4. Replace the components of the gradient and point in the equation from step 3 to obtain the final equation of the tangent plane.

Step by step solution

01

Find the gradient of \(F\)

To find the gradient of \(F\), we need to compute the partial derivatives with respect to \(x\), \(y\), and \(z\). These partial derivatives form the components of the gradient vector: $$\nabla F = \left(\frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z}\right)$$ Evaluate the gradient at the given point \((a, b, c)\).
02

Compute the gradient of \(F\) at \((a, b, c)\)

To compute the gradient of \(F\) at \((a, b, c)\), we substitute these values into the gradient vector: $$\nabla F(a, b, c) = \left(\frac{\partial F}{\partial x}(a, b, c), \frac{\partial F}{\partial y}(a, b, c), \frac{\partial F}{\partial z}(a, b, c)\right)$$ This gives us the normal vector of the tangent plane at \((a, b, c)\).
03

Write the point-normal form of the tangent plane equation

Now that we have the normal vector \(\nabla F(a, b, c)\) and a point \((a, b, c)\) on the plane, we can use the point-normal form of the plane equation to write the equation of the tangent plane: $$\nabla F(a, b, c) \cdot (x - a, y - b, z - c) = 0$$ Where \(\cdot\) denotes the dot product.
04

Write the equation of the tangent plane

To write the equation of the tangent plane, simply substitute the components of \(\nabla F(a, b, c)\) and \((a, b, c)\) into the plane equation from Step 3: $$\left(\frac{\partial F}{\partial x}(a, b, c)\right)(x - a) + \left(\frac{\partial F}{\partial y}(a, b, c)\right)(y - b) + \left(\frac{\partial F}{\partial z}(a, b, c)\right)(z - c) = 0$$ This gives us our final equation of the tangent plane to the surface \(F(x, y, z) = 0\) at the point \((a, b, c)\).

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