91Ó°ÊÓ

The concept of gradient is crucial in vector calculus. It relates to how a scalar field changes in space. The gradient of a scalar field, denoted as \( abla f \), is a vector field that points in the direction of greatest increase of the function. In our exercise, we are given the scalar function \( f(x, y) = x^2 y \). This means we want to find how this function changes in space by calculating its gradient.

To find the gradient \( abla f \), we compute the partial derivatives of \( f \) with respect to each coordinate variable. Here's how it works:

• Partial derivative with respect to \( x \): \( \frac{\partial f}{\partial x} = 2xy \)
• Partial derivative with respect to \( y \): \( \frac{\partial f}{\partial y} = x^2 \)

Thus, the gradient vector field is \( abla f = (2xy, x^2) \). This vector field gives us a sense of direction and magnitude of how the function \( f \) increases in the \( x \) and \( y \) directions.

Divergence is a measure of how much a vector field spreads out from a point. In simple terms, think of it as how much the field 'diverges' or spreads out. For a 2D vector field \( \vec{F} = (P, Q) \), the divergence is calculated using the formula \( abla \cdot \vec{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \). Let's apply it to the vector field \( \vec{F} = (2xy, x^2) \) we previously computed as the gradient.

Here’s how we find the divergence:

• \( \frac{\partial (2xy)}{\partial x} = 2y \)
• \( \frac{\partial (x^2)}{\partial y} = 0 \)

Adding these partial derivatives together, we get \( abla \cdot \vec{F} = 2y \). This result implies that our vector field has divergence that changes linearly with \( y \), meaning that the field spreads out more as \( y \) increases.

Curl is used in vector calculus to measure a field's tendency to rotate around a point. In simpler terms, it looks at the rotational motion imparted by the field. For a 2D vector field \( \vec{F} = (P, Q) \), the curl is expressed as a scalar by the formula \( abla \times \vec{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \). Let's calculate this for our vector field \( \vec{F} = (2xy, x^2) \).

Here’s how we find the curl:

• \( \frac{\partial (x^2)}{\partial x} = 2x \)
• \( \frac{\partial (2xy)}{\partial y} = 2x \)

Subtracting these, we find \( abla \times \vec{F} = 2x - 2x = 0 \). The curl being zero confirms that there is no rotational motion in our field. This characteristic is typical for gradient fields, suggesting that the field is conservative.