Problem 50 - Calculus with Concepts in Calculus

Question

We say that two surfaces are normal at a given point if their tangent planes at that point are perpendicular to one another.Show that the pair of surfaces are normal at the given point. \(x^{2}+y^{2}+z^{2}=16\) and \(z^{2}=x^{2}+y^{2} ;(2,2,2 \sqrt{2})\)

Step-by-Step Solution

Answer

The surfaces are normal at the point because their tangent planes are perpendicular; the dot product of the gradients is zero.

Step 1: Find the Gradient of the First Surface

The first surface is given by the equation \( x^2 + y^2 + z^2 = 16 \). The gradient of this surface, \( abla f \), can be computed as:\[ abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) = (2x, 2y, 2z) \]. At the point \((2, 2, 2\sqrt{2})\), the gradient is \( abla f = (4, 4, 4\sqrt{2}) \).

Step 2: Find the Gradient of the Second Surface

The second surface is given by the equation \( z^2 = x^2 + y^2 \). The gradient of this surface, \( abla g \), can be calculated as:\[ abla g = \left( \frac{\partial g}{\partial x}, \frac{\partial g}{\partial y}, \frac{\partial g}{\partial z} \right) = (2x, 2y, -2z) \]. At the point \((2, 2, 2\sqrt{2})\), the gradient is \( abla g = (4, 4, -4\sqrt{2}) \).

Step 3: Verify Perpendicularity of the Tangent Planes

To check if the tangent planes are perpendicular, we compute the dot product of the gradients \( abla f \) and \( abla g \). The dot product is:\[ abla f \cdot abla g = (4 \cdot 4) + (4 \cdot 4) + (4\sqrt{2} \cdot -4\sqrt{2}) = 16 + 16 - 32 = 0 \].A dot product of zero indicates that the vectors are perpendicular, confirming that the tangent planes are perpendicular.

Key Concepts

Gradient

In calculus, the gradient is a vector that points in the direction of the greatest rate of increase of a function. For a function of three variables, like the surfaces in our exercise, the gradient is calculated as a vector of partial derivatives.

• The gradient of a function \(f(x, y, z)\) is denoted by \( abla f \) and is expressed as \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \).
• This vector gives us vital information about the slope of the surface in various directions.

To compute the gradient at a specific point, substitute the point's coordinates into the gradient vector. For example, in the step by step solution, the gradient of the surface \( x^2 + y^2 + z^2 = 16 \) at the point \((2, 2, 2\sqrt{2})\) results in \( abla f = (4, 4, 4\sqrt{2}) \). Knowing the gradient is fundamental in determining how steep the surface is at any point.

Tangent Planes

The tangent plane to a surface at a given point is an essential concept in multivariable calculus. It represents the plane that "just touches" the surface at that point.

• The direction of the tangent plane is guided by the gradient vector at the point on the surface.
• If you imagine the surface as a terrain, the tangent plane is the imaginary perfectly flat surface that "sits" on the terrain at a point.

Understanding tangent planes helps in visualizing how the function behaves locally and in solving problems like finding points of intersection. In our example, two surfaces are said to be normal at a point if their respective tangent planes at that point are perpendicular to each other.

Perpendicular Vectors

Vectors are considered perpendicular when their dot product equals zero. In geometry and physics, this concept is used to determine the "right angle" relationship between vectors. This is especially important in problems involving angles and orthogonality.

• Perpendicular vectors imply a 90-degree angle between them.
• In three-dimensional space, such vectors define perpendicular planes.

In our exercise, the perpendicularity of tangent planes is proven by showing the dot product of the gradients is zero, confirming they are perpendicular vectors. This perpendicularity indicates that the surfaces touch but do not cross each other at the given point.

Dot Product

The dot product of two vectors is a crucial operation in vector calculus, representing a method to multiply them. The result is a scalar quantity, not a vector.

• For vectors \(\mathbf{a} = (a_1, a_2, a_3)\) and \(\mathbf{b} = (b_1, b_2, b_3)\), the dot product is calculated as \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \).
• It is used to find angles between vectors and determine perpendicularity (orthogonality).

A dot product of zero indicates that the vectors are perpendicular. In our solution, the zero dot product of the gradients confirms that the tangent planes of the surfaces are perpendicular to each other.