Problem 1 - Calculus: Early Transcendental Functions

Question

Find the curl and divergence of the given vector field. $$x^{2} \mathbf{i}-3 x y \mathbf{j}$$

Step-by-Step Solution

Answer

Based on the computations, the curl of the vector field is \(0\mathbf{k}\) and the divergence of the vector field is \(2x - 3y\).

Step 1: Calculate the partial derivatives necessary

Calculate the following partial derivatives of the components of the vector field which are necessary for the curl and divergence formulas: \[ \frac{\partial}{\partial x} (x^{2})\], \[ \frac{\partial}{\partial y} (-3xy)\].

Step 2: Calculate the curl of the vector field

The curl of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} \) is given by \[Curl(\mathbf{F}) = \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\mathbf{k}\]. So, calculate \[Curl(\mathbf{F}) = \left( \frac{\partial (-3xy)}{\partial x} - \frac{\partial x^{2}}{\partial y}\right)\mathbf{k}\].

Step 3: Calculate the divergence of the vector field

The divergence of a vector field \(F = P\mathbf{i} + Q\mathbf{j}\) is given by \[div(\mathbf{F}) = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}\]. So, calculate \[div(\mathbf{F}) = \frac{\partial x^{2}}{\partial x} + \frac{\partial (-3xy)}{\partial y}\].

Key Concepts

Curl

When we talk about the curl of a vector field, we're exploring its rotational effect around a point in space. Imagine a whirlpool or a tornado; these circular motions are similar to what the curl represents mathematically. In vector calculus, the curl of a two-dimensional vector field, like the one given \[ \mathbf{F} = x^2 \mathbf{i} - 3xy \mathbf{j} \], is calculated in the direction of the \( \mathbf{k} \) component, which is perpendicular to the \( \mathbf{i} \) and \( \mathbf{j} \) components. To understand the formula \[ \text{Curl}(\mathbf{F}) = \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \mathbf{k} \], you should notice two important aspects:

• Partial Derivative of Q with respect to x: This tells us about the rate of change of the component Q (\( -3xy \) in our case) in the direction of x, contributing to the tendency to rotate.
• Partial Derivative of P with respect to y: This represents how the component P (\( x^2 \)) changes in the y direction, also impacting the rotation pattern.

The difference between these derivatives gives us the resulting rotational motion or 'curl'. In practical problems, a zero curl means there is no rotation, like still water, while a non-zero curl signifies some rotational effect.

Divergence

Divergence might sound complex, but it's simply a way to determine if a vector field is acting like a source or a sink around a point. It measures how much a given vector field spreads outwardly or shrinks inwardly around a given point. Like water flowing from a tap or into a drain.To compute the divergence of our vector field \[ \mathbf{F} = x^2 \mathbf{i} - 3xy \mathbf{j} \], we use the formula: \[ \text{div}(\mathbf{F}) = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \]. This involves:

• Partial Derivative of P with respect to x: Here, P is \( x^2 \), and we're finding out how much this component stretches or compresses along the x-axis.
• Partial Derivative of Q with respect to y: For Q which is \( -3xy \), it involves noticing the change in the y-axis direction.

By summing these partials, divergence gives us a scalar value. A positive value means the field sources or spreads out, while a negative value indicates it sinks or converges. If the divergence is zero in some regions, it suggests neither behavior predominantly, often implying conservation-like properties within the field.

Partial Derivatives

Partial derivatives form the building block of calculating curl and divergence in vector fields. These derivatives are like zooming in on specific parts of a multivariable function, showing how certain variables cause changes while keeping others constant.In our vector field \[ \mathbf{F} = x^2 \mathbf{i} - 3xy \mathbf{j} \], each component is treated like a function of both x and y. For example:

• The partial derivative of \( x^2 \) with respect to x, \( \frac{\partial}{\partial x}(x^2) = 2x \), gives us the rate of change of \( x^2 \) as we move along the x-axis, as if y is fixed.
• The partial derivative of \( -3xy \) with respect to y, \( \frac{\partial}{\partial y}(-3xy) = -3x \), measures how \( -3xy \) changes along the y-axis, assuming x enjoys a constant value.

By grasping partial derivatives, you can effectively predict how each part of the vector field reacts locally, offering a clearer picture of the complete vector environment. They're essential in identifying the contributions of each variable to the overall field behavior, be it rotational or expansive.