Problem 45 - Calculus Early Transcendentals

Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The planes tangent to the cylinder \(x^{2}+z^{2}=1\) in \(\mathbb{R}^{3}\) all have the form \(a x+b z+c=0\) b. Suppose \(w=x y / z,\) for \(x>0, y>0,\) and \(z>0 .\) A decrease in \(z\) with \(x\) and \(y\) fixed results in an increase in \(w\) c. The gradient \(\nabla F(a, b, c)\) lies in the plane tangent to the surface \(F(x, y, z)=0\) at \((a, b, c)\)

Step-by-Step Solution

Answer

#Short Answer# a. The statement is true because the tangent planes to the cylinder have the form \(ax + cz - 1 = 0\), which matches the given form \(ax + bz + c = 0\) (with \(b = 0\) and \(c = -1\)). b. The statement is true because decreasing \(z\) when \(x\) and \(y\) are fixed increases the value of \(w = xy/z\). c. The statement is false because the gradient vector is orthogonal (perpendicular) to the tangent plane, not lying within the plane itself.

Step 1: Find the gradient of the cylinder function

First, let's define a function representing the cylinder: \(F(x, z) = x^2 + z^2 - 1\). Now, we can find the gradient of this function, denoted as \(\nabla F\). \(\nabla F(x, z) = \left(\frac{\partial F}{\partial x}, \frac{\partial F}{\partial z}\right) = (2x, 2z)\)

Step 2: Determine the equation of the tangent planes

For the cylinder, we have \(\nabla F \cdot (x - a, z - c) = 0\), where \((a, c)\) is a point on the cylinder. The dot product of the two vectors gives us: \((2a)(x - a) + (2c)(z - c) = 0\). Simplifying, we get: \(ax + cz - (a^2 + c^2) = 0\). Since \(a^2 + c^2 = 1\), we find the equation of the tangent planes to be: \(ax + cz - 1 = 0\). This matches the given form \(ax + bz + c = 0\) (with \(b = 0\) and \(c = -1\)), so the statement is true. b. Decreasing \(z\) with \(x\) and \(y\) fixed

Step 1: Analyze the effect of decreasing \(z\)

We are given \(w = xy/z\). Let's analyze the effect of decreasing \(z\) on the value of \(w\). We know that \(x>0\), \(y>0\), and \(z>0\). If we decrease \(z\), its denominator will become smaller, which increases the value of the fraction. Therefore, a decrease in \(z\) will result in an increase in \(w\). So, the statement is true. c. Gradient and tangent plane

Step 1: Examine the relation between the gradient and the tangent plane

We know that the tangent plane to the surface \(F(x, y, z) = 0\) at a point \((a, b, c)\) is given by the equation \(\nabla F(a, b, c) \cdot (x - a, y - b, z - c) = 0\). This equation states that the dot product of the gradient vector and the vector \((x - a, y - b, z - c)\) is zero, which means that these vectors are orthogonal (perpendicular) to each other. Since the gradient vector is perpendicular to the tangent plane, it lies in the plane's normal direction. The statement is false since the gradient vector is orthogonal to the tangent plane, not lying within the plane itself.

Key Concepts

Gradient

In vector calculus, the gradient is a vector that contains the partial derivatives of a function with respect to its variables. If you have a function, for example, \( F(x, z) = x^2 + z^2 - 1 \), the gradient \( abla F \) would be computed as \( abla F(x, z) = \left(\frac{\partial F}{\partial x}, \frac{\partial F}{\partial z}\right) = (2x, 2z) \).
This vector points in the direction of the steepest increase of the function. Hence, for a function of two variables, the gradient lets us know which way to move in \( x \) and \( z \) to increase the function as quickly as possible.

• Each component of the gradient vector is a partial derivative with respect to the respective variable.
• The magnitude of the gradient gives the rate of increase at a specific point.

Recognizing the gradient is crucial for understanding how surfaces behave in three-dimensional space.

Tangent Planes

A tangent plane is a plane that touches a surface at exactly one point and is "parallel" to the surface's immediate slope. For a surface defined by an equation like \( x^2 + z^2 = 1 \), the tangent plane at a point \((a, c)\) on the surface can be determined using the gradient.
The equation of the tangent plane is derived from the gradient by calculating \( abla F \cdot (x - a, z - c) = 0 \). Simplifying this equation gives us \( ax + cz - 1 = 0 \), which describes the tangent plane.

• The coefficients \( a \) and \( c \) are components of the gradient at the point \((a, c)\).
• The constant term arises from the condition \( a^2 + c^2 = 1 \), ensuring the plane is tangent at the cylinder.

This concept is pivotal for determining how surfaces like cylinders align with planes in three dimensions.

Cylinders

Cylinders in mathematics are simple geometrical objects, but their spatial properties make them crucial in calculus. A cylinder like \( x^2 + z^2 = 1 \) in 3D space is a circular cylinder with its axis along the y-axis.
When dealing with cylinders, understanding cross-sections and projections is essential. Here, each cross-section parallel to the \(xz\)-plane is a circle.

• The defining equation \( x^2 + z^2 = 1 \) describes all the 2D circles at any fixed \( y \)-value on the cylinder.
• Each circular cross-section has a radius of 1, hence why the equation sums to 1.

Cylinders are perfect examples for applying concepts like tangent planes and gradients since their symmetry makes calculations more straightforward.

Orthogonality

Orthogonality in vector calculus refers to perpendicular vectors. When two vectors are orthogonal, their dot product is zero. This concept is crucial in understanding gradients and tangent planes.
Take the gradient \( abla F \) from a point on a surface. Now consider the tangent plane at that point. The two are orthogonal to each other according to the equation \( abla F(a, b, c) \cdot (x - a, y - b, z - c) = 0 \).

• The dot product equaling zero implies that the gradient is perpendicular to the tangent plane.
• This perpendicular relationship helps in finding normals to surfaces and characterizing the direction and orientation of surfaces in space.

Understanding orthogonality assists in picturing how vectors and surfaces interact in a spatial context.