Question

Find the divergence and curl of the given vector field. $$ \vec{F}=\left\langle-y^{2}, x\right\rangle $$

Short answer

Divergence is 0; Curl is \( (1 + 2y) \hat{k} \).

Step-by-step solution

Define the Divergence

The divergence of a vector field \( \vec{F} = \langle F_1, F_2 \rangle \) is defined as \( abla \cdot \vec{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} \). For the vector field \( \vec{F} = \langle -y^2, x \rangle \), we have \( F_1 = -y^2 \) and \( F_2 = x \).

Compute Partial Derivatives for Divergence

Calculate the partial derivatives:- \( \frac{\partial F_1}{\partial x} = \frac{\partial}{\partial x}(-y^2) = 0 \) (since \(-y^2\) has no \(x\) dependence), - \( \frac{\partial F_2}{\partial y} = \frac{\partial}{\partial y}(x) = 0 \) (since \(x\) has no \(y\) dependence).

Calculate Divergence

Using the results from Step 2, the divergence is \( abla \cdot \vec{F} = 0 + 0 = 0 \). The divergence of the vector field is 0.

Define the Curl

In two dimensions, the curl of a vector field \( \vec{F} = \langle F_1, F_2 \rangle \) is defined as \( abla \times \vec{F} = \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \hat{k} \).

Compute Partial Derivatives for Curl

Calculate the partial derivatives needed for the curl:- \( \frac{\partial F_2}{\partial x} = \frac{\partial x}{\partial x} = 1 \),- \( \frac{\partial F_1}{\partial y} = \frac{\partial}{\partial y}(-y^2) = -2y \).

Calculate Curl

Using the results from Step 5, the curl is \( abla \times \vec{F} = \left( 1 + 2y \right) \hat{k} \). So, the curl of the vector field is \( (1 + 2y) \hat{k} \).

Key concepts

Divergence

In vector calculus, divergence is a crucial concept that describes the behavior of a vector field. Specifically, it measures the magnitude of a field's source or sink at a given point. Consider the vector field \( \vec{F} = \langle F_1, F_2 \rangle \), the divergence is calculated as \( abla \cdot \vec{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} \).

For the field \( \vec{F} = \langle -y^2, x \rangle \), we found the divergence is zero. This result indicates that the vector field is neither expanding nor compressing at any point.

• The physical interpretation could be related to fluid flow - a zero divergence means the flow is constant, no net flow in or out at any point.
• Applications of divergence can be found in physics, such as in electromagnetism, where it helps determine the presence of charges.

Curl

Curl is another important part of vector calculus. It measures the rotational motion or the twisting tendency at a point in the vector field. For a 2D vector field \( \vec{F} = \langle F_1, F_2 \rangle \), curl is given by \( abla \times \vec{F} = \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \hat{k} \).

When we apply this to our vector field \( \vec{F} = \langle -y^2, x \rangle \), the curl becomes \( (1 + 2y) \hat{k} \). This non-zero result indicates that the vector field has rotational motion, and the rate changes depending on the \( y \) coordinate.

• Curl is useful in fields like fluid dynamics and electromagnetism, where rotation or circulation of the field is involved.
• The presence of curl suggests a swirling motion, similar to how water swirls around a drain.

Vector Field

A vector field is an assignment of a vector to each point in a subset of space. In two dimensions, it's often represented as \( \vec{F} = \langle F_1(x, y), F_2(x, y) \rangle \). Think of it as a multitude of arrows spread across a region, where each arrow corresponds to a vector at each position.

The problem provides \( \vec{F} = \langle -y^2, x \rangle \), depicting how vectors change across the \( x \)-\( y \) plane. Vector fields have different applications depending on the context, such as:

• Describing the velocity of a moving fluid across space.
• Illustrating gravitational fields as force vectors around massive objects.

Understanding vector fields is critical in many areas of science and engineering where directional quantities, like force or velocity, vary across space.

Partial Derivatives

Partial derivatives are at the heart of vector calculus. They provide a way to understand how functions change when one variable changes and others remain constant. When dealing with multi-variable functions like in vector calculus, partial derivatives are used.

In our exercise, we computed partial derivatives to find divergence and curl. For instance, \( \frac{\partial F_1}{\partial x} \) for \( F_1 = -y^2 \) evaluates to 0 since \( F_1 \) doesn't depend on \( x \). Similarly, \( \frac{\partial F_2}{\partial y} \) for \( F_2 = x \) also yields 0.

• Understanding partial derivatives is essential for calculating and interpreting concepts like divergence and curl.
• They allow us to analyze how functions behave when changing one variable while keeping others fixed.

Partial derivatives expand the capability of calculus to explore complex scenarios, which is indispensable in mathematical modeling and problem-solving.