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Chapter 13: Problem 11

Determine whether or not the vector field is conserva- tive. If it is conservative, find a function \(f\) such that \(\mathbf{F}=\nabla f\) $$\mathbf{F}(x, y, z)=y^{2} z^{3} \mathbf{i}+2 x y z^{3} \mathbf{j}+3 x y^{2} z^{2} \mathbf{k}$$

Short Answer

Expert verified
The vector field is conservative with the potential function \( f(x,y,z) = x y^2 z^3 + C \).

Step by step solution

01

Check if the Vector Field is Conservative

For a vector field \( \mathbf{F} = P(x, y, z) \mathbf{i} + Q(x, y, z) \mathbf{j} + R(x, y, z) \mathbf{k} \) to be conservative, it must be the gradient of some scalar function \( f \). By definition, \( \mathbf{F} \) is conservative if \[ abla \times \mathbf{F} = \mathbf{0} \] First, compute the partial derivatives involved in checking the curl:- \( P = y^2 z^3 \)- \( Q = 2xyz^3 \)- \( R = 3xy^2z^2 \)Check for curl components:- \( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} = \frac{\partial (3xy^2z^2)}{\partial y} - \frac{\partial (2xyz^3)}{\partial z} = 6xyz^2 - 6xyz^2 = 0 \)- \( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} = \frac{\partial (y^2z^3)}{\partial z} - \frac{\partial (3xy^2z^2)}{\partial x} = 3y^2z^2 - 3y^2z^2 = 0 \)- \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = \frac{\partial (2xyz^3)}{\partial x} - \frac{\partial (y^2z^3)}{\partial y} = 2yz^3 - 2yz^3 = 0 \)All components are zero, so \( abla \times \mathbf{F} = \mathbf{0} \). The vector field is conservative.
02

Find the Potential Function \(f\)

Since \( \mathbf{F} = abla f \), we need to integrate each component to find \( f(x, y, z) \).1. **Integrate \( P \) with respect to \( x \):** \[ \int y^2 z^3 \, dx = x y^2 z^3 + g(y, z) \] Here, \( g(y, z) \) is an arbitrary function of \( y \) and \( z \), since differentiating \( f \) with respect to \( x \) should yield a result that matches the given \( P \).2. **Use the \( Q \) component to determine \( g(y,z) \):** \[ \frac{\partial f}{\partial y} = 2xyz^3 = 2xyz^3 + \frac{\partial g(y, z)}{\partial y} \] Since the terms involving \( x \) match, we identify: \[ \frac{\partial g(y, z)}{\partial y} = 0 \Rightarrow g(y, z) = h(z) \] \( h(z) \) is an arbitrary function of \( z \).3. **Use \( R \) to find \( h(z) \):** \[ \frac{\partial f}{\partial z} = 3xy^2z^2 \] \[ \frac{\partial (x y^2 z^3 + h(z))}{\partial z} = 3xy^2z^2 + \frac{dh(z)}{dz} \] Matching terms gives: \[ \frac{dh(z)}{dz} = 0 \Rightarrow h(z) = C \] \( C \) is a constant.4. **Combine results to get \( f(x,y,z) \):** \[ f(x, y, z) = x y^2 z^3 + C \] Thus, the potential function is \( f(x, y, z) = x y^2 z^3 + C \), with \( C \) as a constant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gradient
The concept of a "gradient" is fundamental in vector calculus. It represents how a function changes at any point in space. Think of the gradient as a vector that points in the direction of the steepest ascent of the function.

The gradient of a scalar function, say \( f(x, y, z) \), is denoted by \( abla f \) and is defined as a vector composed of partial derivatives:
  • \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \)
In simplicity, it tells us the direction and rate at which the function \( f \) increases the fastest.

For a vector field \( \mathbf{F} \) to be conservative, it must be the gradient of some potential function \( f \). This means if we can express \( \mathbf{F} \) as \( abla f \), then \( \mathbf{F} \) is conservative. Often, solving for \( f \) involves integrating the components of \( \mathbf{F} \), as demonstrated in the solution where \( f(x, y, z) = x y^2 z^3 + C \). This potential function allows us to fully describe the behavior of \( \mathbf{F} \).
Curl of a Vector Field
The "curl" of a vector field is a measure of the field's tendency to rotate around a point. It's like the rotational speed of a tiny paddle wheel placed in the field. Mathematically, it is denoted as \( abla \times \mathbf{F} \).

Given a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \), the curl \( abla \times \mathbf{F} \) is calculated as:
  • \( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \)
  • \( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \)
  • \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \)

If the curl is zero, the vector field has no "twist" at any point, which is a necessary condition for a field to be conservative. In our example, all these partial derivatives differences were zero, indicating that \( abla \times \mathbf{F} = \mathbf{0} \). Hence, the field is conservative, meaning it can be expressed as the gradient of some potential function \( f \). This absence of curl simplifies many calculations and offers insight into the field's behavior.
Potential Function
A "potential function" \( f \) is a scalar function from which a vector field \( \mathbf{F} \) can be derived by taking the gradient. It encapsulates all the information about the vector field in a single function.

To find a potential function for a vector field \( \mathbf{F} \), one typically integrates the field's components. This process is shown in the step-by-step solution where integration was performed on each component part of \( \mathbf{F} \):

  • Integrate the first component with respect to its variable while treating others as constants.
  • Match the other components through integration to determine any leftover functions.
  • Finally, assemble these pieces to form \( f(x,y,z) \).

The presence of a potential function simplifies problem-solving, as it can be used to calculate work done by the field. In this specific example, the potential function \( f(x, y, z) = x y^2 z^3 + C \) allows us to easily understand how the field operates throughout space. Knowing a potential function exists ensures the field is free of closed loops and adheres to the path-independent property.

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Most popular questions from this chapter

(a) Use Stokes' Theorem to evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r},\) where \(\mathbf{F}(x, y, z)=x^{2} y \mathbf{i}+\frac{1}{3} x^{3} \mathbf{j}+x y \mathbf{k}\) and \(C\) is the curve of intersection of the hyperbolic paraboloid \(z=y^{2}-x^{2}\) and the cylinder \(x^{2}+y^{2}=1\) oriented counterclockwise as viewed from above. (b) Graph both the hyperbolic paraboloid and the cylinder with domains chosen so that you can see the curve \(C\) and the surface that you used in part (a). (c) Find parametric equations for \(C\) and use them to graph \(C .\)

\(1-4\) Verify that the Divergence Theorem is true for the vector field \(F\) on the region \(E .\) $$\mathbf{F}(x, y, z)=\left\langle x^{2},-y, z\right\rangle$$ \(E\) is the solid cylinder \(y^{2}+z^{2} \leqslant 9,0 \leqslant x \leqslant 2\)

Find the mass of a thin funnel in the shape of a cone \(z=\sqrt{x^{2}+y^{2}}, 1 \leqslant z \leqslant 4,\) if its density function is \(\rho(x, y, z)=10-z\).

Evaluate the surface integral \(\iint_{s} \mathbf{F} \cdot d \mathbf{S}\) for the given vector field \(\mathbf{F}\) and the oriented surface \(S .\) In other words, find the flux of \(\mathbf{F}\) across \(S .\) For closed surfaces, use the positive (outward) orientation. $$\begin{array}{l}{\mathbf{F}(x, y, z)=x y \mathbf{i}+4 x^{2} \mathbf{j}+y z \mathbf{k}, \quad S \text { is the surface } z=x e^{y}} \\ {0 \leqslant x \leqslant 1,0 \leqslant y \leqslant 1, \text { with upward orientation }}\end{array}$$

\(21-27=\) Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If \(f\) is a scalar field and \(\mathbf{F},\) \(\mathbf{G}\) are vector fields, then \(f \mathbf{F}, \mathbf{F} \cdot \mathbf{G},\) and \(\mathbf{F} \times \mathbf{G}\) are defined by \((f \mathbf{F})(x, y, z)=f(x, y, z) \mathbf{F}(x, y, z)\) \((\mathbf{F} \cdot \mathbf{G})(x, y, z)=\mathbf{F}(x, y, z) \cdot \mathbf{G}(x, y, z)\) \((\mathbf{F} \times \mathbf{G})(x, y, z)=\mathbf{F}(x, y, z) \times \mathbf{G}(x, y, z)\) $$\operatorname{div}(\mathbf{F} \times \mathbf{G})=\mathbf{G} \cdot \operatorname{curl} \mathbf{F}-\mathbf{F} \cdot \operatorname{curl} \mathbf{G}$$
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