Problem 44 - Thomas Calculus

Question

Two "central" fields Find a field \(\mathbf{F}=M(x, y) \mathbf{i}+N(x, y) \mathbf{j}\) in the \(x y\) -plane with the property that at each point \((x, y) \neq(0,0), \mathbf{F}\) points toward the origin and \(|\mathbf{F}|\) is (a) the distance from \((x, y)\) to the origin, (b) inversely proportional to the distance from \((x, y)\) to the origin. (The field is undefined at \((0,0)\) .)

Step-by-Step Solution

Answer

(a) \( \mathbf{F} = -(x\mathbf{i} + y\mathbf{j}) \); (b) \( \mathbf{F} = -\frac{c}{x^2 + y^2}(x\mathbf{i} + y\mathbf{j}) \).

Step 1: Understanding the Problem

We need to find a vector field \( \mathbf{F} = M(x, y) \mathbf{i} + N(x, y) \mathbf{j} \) such that it points towards the origin at each point \((x, y) eq (0,0)\). This means that the field's direction should always be towards \((0,0)\), which implies that the field vector at each point should be a negative multiple of the position vector \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} \).

Step 2: Express \( \mathbf{F} \) in Terms of \( \mathbf{r} \)

Given that \( \mathbf{F} \) should point towards the origin, we can express it as \( \mathbf{F} = -k(x, y) (x\mathbf{i} + y\mathbf{j}) \), where \( k(x, y) \) is a function that determines the magnitude of \( \mathbf{F} \) according to the specific conditions given in part (a) or (b).

Step 3: Solve for Part (a)

For part (a), \( |\mathbf{F}| \) is the distance from \((x, y)\) to the origin, which is \( \sqrt{x^2 + y^2} \). Thus:\[|\mathbf{F}| = k(x, y) \sqrt{x^2 + y^2} = \sqrt{x^2 + y^2}\]This implies \( k(x, y) = 1 \). Therefore:\[\mathbf{F} = -(x\mathbf{i} + y\mathbf{j})\]

Step 4: Solve for Part (b)

For part (b), \( |\mathbf{F}| \) is inversely proportional to the distance from \((x, y)\) to the origin. This means:\[|\mathbf{F}| = \frac{c}{\sqrt{x^2 + y^2}}\]where \( c \) is a constant. Thus:\[k(x, y) \sqrt{x^2 + y^2} = \frac{c}{\sqrt{x^2 + y^2}}\]Solving for \( k(x, y) \), we have \( k(x, y) = \frac{c}{(x^2 + y^2)} \). Therefore:\[\mathbf{F} = -\frac{c}{x^2 + y^2}(x\mathbf{i} + y\mathbf{j})\]

Key Concepts

Central Fields

Central fields are fascinating aspects of vector fields. They are specially designed fields where the direction vectors always point towards a central point, often the origin in a coordinate system. In simpler terms, no matter where you are in the plane, the vector field always "pulls" towards that central point. This unique characteristic makes central fields very important in physics, especially in problems involving forces like gravity or electric fields.
To better understand the core idea, consider the following features:

• Central fields are radially symmetric, meaning their influence spreads evenly from a central point outward in all directions.
• The strength and direction of the vectors in these fields depend on both their distance and the angle relative to the origin.

Central fields can sometimes represent physical phenomena like gravitational attraction, where every particle moves directed towards a mass at the origin. This concept is crucial for understanding how larger systems function and interact with central forces.

Inverse Proportionality

The idea of inverse proportionality is an essential concept in mathematics and physics, especially when dealing with vector fields. Simply put, when two quantities are inversely proportional, as one quantity increases, the other decreases. The precision of this relationship can be captured in mathematical form: \[ y \propto \frac{1}{x} \]This formula indicates that "y" is inversely proportional to "x".
In the context of the vector field problem, the magnitude of the field vectors is designed to decrease as they move away from the origin. This specific form of inverse proportionality is often observed in gravitational and electrostatic fields where the force experienced by a particle decreases with an increase in distance from the source.
Key aspects include:

• The field strength will decrease as the distance from the origin increases.
• This behavior ensures that objects experience less force the further they are from the center.

Inverse proportionality ties in with the field's nature of pulling objects toward a center, balancing the force's reach across different distances.

Position Vectors

Position vectors play a pivotal role in describing locations within a vector field. A position vector essentially indicates the position of a point in space relative to the origin of the coordinate system. For example, the position vector \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} \) precisely describes any point \((x,y)\) in a two-dimensional plane.
The significance of position vectors in our context involves their use as a reference for determining the direction of the vector field. Given that the field should point toward the origin, position vectors help us establish the necessary direction by providing a clear path "backward" along their lines:

• Position vectors start at the origin and point outward towards a specified point.
• In our vector field problem, the field vectors are oriented in the opposite direction of these position vectors. This ensures that the entire field consistently points back to the origin.

Understanding position vectors is crucial for visualizing and calculating how a vector field operates throughout a given space. Mastery of this concept enhances one's ability to engage with complex systems modeled by vector fields efficiently.