Find expert answers and community support for all your questions on IDNLearn.com. Our experts provide timely and accurate responses to help you navigate any topic or issue with confidence.
Sagot :
To solve the problem using the Fundamental Theorem of Line Integrals, we need to proceed through the following steps:
1. Verify if the vector field [tex]\(\vec{F}\)[/tex] is conservative.
2. Find the potential function [tex]\(f(x, y, z)\)[/tex] if the vector field is conservative.
3. Evaluate the potential function at the endpoints [tex]\((1, 0, 2)\)[/tex] and [tex]\((1, 1, 1)\)[/tex].
4. Calculate the line integral using the fundamental theorem for the conservative vector field.
---
### Step 1: Verify if the Vector Field is Conservative
A vector field [tex]\(\vec{F}(x, y, z) = \langle P, Q, R \rangle\)[/tex] is conservative if there exists a potential function [tex]\(f(x, y, z)\)[/tex] such that:
[tex]\[ \vec{F} = \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \][/tex]
One way to check if [tex]\(\vec{F}\)[/tex] is conservative is to verify that the curl of [tex]\(\vec{F}\)[/tex] is zero:
[tex]\[ \nabla \times \vec{F} = \vec{0} \][/tex]
Given:
[tex]\[ \vec{F}(x, y, z) = \langle 4x - 2y + 2z, -2x + 2yz + z^2, 2x + 2yz + y^2 \rangle \][/tex]
We compute the curl [tex]\(\nabla \times \vec{F}\)[/tex]:
[tex]\[ \nabla \times \vec{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \][/tex]
Calculate each component:
1. [tex]\(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\)[/tex]:
[tex]\[ \frac{\partial}{\partial y} (2x + 2yz + y^2) = 2z + 2y \][/tex]
[tex]\[ \frac{\partial}{\partial z} (-2x + 2yz + z^2) = 2y + 2z \][/tex]
[tex]\[ \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} = (2z + 2y) - (2y + 2z) = 0 \][/tex]
2. [tex]\(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\)[/tex]:
[tex]\[ \frac{\partial}{\partial z} (4x - 2y + 2z) = 2 \][/tex]
[tex]\[ \frac{\partial}{\partial x} (2x + 2yz + y^2) = 2 \][/tex]
[tex]\[ \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} = 2 - 2 = 0 \][/tex]
3. [tex]\(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\)[/tex]:
[tex]\[ \frac{\partial}{\partial x} (-2x + 2yz + z^2) = -2 \][/tex]
[tex]\[ \frac{\partial}{\partial y} (4x - 2y + 2z) = -2 \][/tex]
[tex]\[ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = -2 - (-2) = 0 \][/tex]
Since all components of the curl are zero, [tex]\(\nabla \times \vec{F} = \langle 0, 0, 0 \rangle\)[/tex]. Thus, [tex]\(\vec{F}\)[/tex] is conservative.
---
### Step 2: Find the Potential Function [tex]\(f(x, y, z)\)[/tex]
Since [tex]\(\vec{F}\)[/tex] is conservative, there exists a potential function [tex]\(f(x, y, z)\)[/tex] such that:
[tex]\[ \vec{F} = \nabla f \][/tex]
We need to find [tex]\(f\)[/tex] such that:
[tex]\[ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) = \left( 4x - 2y + 2z, -2x + 2yz + z^2, 2x + 2yz + y^2 \right) \][/tex]
Integrate the first component with respect to [tex]\(x\)[/tex]:
[tex]\[ \frac{\partial f}{\partial x} = 4x - 2y + 2z \implies f(x, y, z) = 2x^2 - 2xy + 2xz + g(y, z) \][/tex]
Here, [tex]\(g(y, z)\)[/tex] is a function of [tex]\(y\)[/tex] and [tex]\(z\)[/tex] because the derivative with respect to [tex]\(x\)[/tex] should not change [tex]\(g(y, z)\)[/tex].
Integrate the second component with respect to [tex]\(y\)[/tex]:
[tex]\[ \frac{\partial f}{\partial y} = -2x + 2yz + z^2 \implies f(x, y, z) = -2xy + y^2z + z^2y + h(x, z) \][/tex]
Combining and resolving the terms, we assume consistent forms:
[tex]\[ f(x, y, z) = 2x^2 - 2xy + 2xz + y^2z + z^2y \][/tex]
Thus, potential function [tex]\( f \)[/tex] is:
[tex]\[ f(x, y, z) = 2x^2 - 2xy + 2xz + y^2z + z^2y \][/tex]
---
### Step 3: Evaluate the Potential Function at the Endpoints
Evaluate [tex]\( f \)[/tex] at [tex]\((1, 0, 2)\)[/tex]:
[tex]\[ f(1, 0, 2) = 2(1)^2 - 2(1)(0) + 2(1)(2) + (0)^2(2) + (2)^2(0) = 2 + 4= 6 \][/tex]
Evaluate [tex]\( f \)[/tex] at [tex]\((1, 1, 1)\)[/tex]:
[tex]\[ f(1, 1, 1) = 2(1)^2 - 2(1)(1) + 2(1)(1) + (1)^2(1) + (1)^2(1) = 2-2+2 +1+1 = 4 \][/tex]
---
### Step 4: Calculate the Line Integral
Using the Fundamental Theorem of Line Integrals:
[tex]\[ \int_C \vec{F} \cdot d\vec{r} = f(1, 1, 1) - f(1, 0, 2) \][/tex]
Thus,
[tex]\[ \int_C \vec{F} \cdot d\vec{r} = 4 - 6 = -2 \][/tex]
Final Answer:
[tex]\[ \boxed{-2} \][/tex]
1. Verify if the vector field [tex]\(\vec{F}\)[/tex] is conservative.
2. Find the potential function [tex]\(f(x, y, z)\)[/tex] if the vector field is conservative.
3. Evaluate the potential function at the endpoints [tex]\((1, 0, 2)\)[/tex] and [tex]\((1, 1, 1)\)[/tex].
4. Calculate the line integral using the fundamental theorem for the conservative vector field.
---
### Step 1: Verify if the Vector Field is Conservative
A vector field [tex]\(\vec{F}(x, y, z) = \langle P, Q, R \rangle\)[/tex] is conservative if there exists a potential function [tex]\(f(x, y, z)\)[/tex] such that:
[tex]\[ \vec{F} = \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \][/tex]
One way to check if [tex]\(\vec{F}\)[/tex] is conservative is to verify that the curl of [tex]\(\vec{F}\)[/tex] is zero:
[tex]\[ \nabla \times \vec{F} = \vec{0} \][/tex]
Given:
[tex]\[ \vec{F}(x, y, z) = \langle 4x - 2y + 2z, -2x + 2yz + z^2, 2x + 2yz + y^2 \rangle \][/tex]
We compute the curl [tex]\(\nabla \times \vec{F}\)[/tex]:
[tex]\[ \nabla \times \vec{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \][/tex]
Calculate each component:
1. [tex]\(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\)[/tex]:
[tex]\[ \frac{\partial}{\partial y} (2x + 2yz + y^2) = 2z + 2y \][/tex]
[tex]\[ \frac{\partial}{\partial z} (-2x + 2yz + z^2) = 2y + 2z \][/tex]
[tex]\[ \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} = (2z + 2y) - (2y + 2z) = 0 \][/tex]
2. [tex]\(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\)[/tex]:
[tex]\[ \frac{\partial}{\partial z} (4x - 2y + 2z) = 2 \][/tex]
[tex]\[ \frac{\partial}{\partial x} (2x + 2yz + y^2) = 2 \][/tex]
[tex]\[ \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} = 2 - 2 = 0 \][/tex]
3. [tex]\(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\)[/tex]:
[tex]\[ \frac{\partial}{\partial x} (-2x + 2yz + z^2) = -2 \][/tex]
[tex]\[ \frac{\partial}{\partial y} (4x - 2y + 2z) = -2 \][/tex]
[tex]\[ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = -2 - (-2) = 0 \][/tex]
Since all components of the curl are zero, [tex]\(\nabla \times \vec{F} = \langle 0, 0, 0 \rangle\)[/tex]. Thus, [tex]\(\vec{F}\)[/tex] is conservative.
---
### Step 2: Find the Potential Function [tex]\(f(x, y, z)\)[/tex]
Since [tex]\(\vec{F}\)[/tex] is conservative, there exists a potential function [tex]\(f(x, y, z)\)[/tex] such that:
[tex]\[ \vec{F} = \nabla f \][/tex]
We need to find [tex]\(f\)[/tex] such that:
[tex]\[ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) = \left( 4x - 2y + 2z, -2x + 2yz + z^2, 2x + 2yz + y^2 \right) \][/tex]
Integrate the first component with respect to [tex]\(x\)[/tex]:
[tex]\[ \frac{\partial f}{\partial x} = 4x - 2y + 2z \implies f(x, y, z) = 2x^2 - 2xy + 2xz + g(y, z) \][/tex]
Here, [tex]\(g(y, z)\)[/tex] is a function of [tex]\(y\)[/tex] and [tex]\(z\)[/tex] because the derivative with respect to [tex]\(x\)[/tex] should not change [tex]\(g(y, z)\)[/tex].
Integrate the second component with respect to [tex]\(y\)[/tex]:
[tex]\[ \frac{\partial f}{\partial y} = -2x + 2yz + z^2 \implies f(x, y, z) = -2xy + y^2z + z^2y + h(x, z) \][/tex]
Combining and resolving the terms, we assume consistent forms:
[tex]\[ f(x, y, z) = 2x^2 - 2xy + 2xz + y^2z + z^2y \][/tex]
Thus, potential function [tex]\( f \)[/tex] is:
[tex]\[ f(x, y, z) = 2x^2 - 2xy + 2xz + y^2z + z^2y \][/tex]
---
### Step 3: Evaluate the Potential Function at the Endpoints
Evaluate [tex]\( f \)[/tex] at [tex]\((1, 0, 2)\)[/tex]:
[tex]\[ f(1, 0, 2) = 2(1)^2 - 2(1)(0) + 2(1)(2) + (0)^2(2) + (2)^2(0) = 2 + 4= 6 \][/tex]
Evaluate [tex]\( f \)[/tex] at [tex]\((1, 1, 1)\)[/tex]:
[tex]\[ f(1, 1, 1) = 2(1)^2 - 2(1)(1) + 2(1)(1) + (1)^2(1) + (1)^2(1) = 2-2+2 +1+1 = 4 \][/tex]
---
### Step 4: Calculate the Line Integral
Using the Fundamental Theorem of Line Integrals:
[tex]\[ \int_C \vec{F} \cdot d\vec{r} = f(1, 1, 1) - f(1, 0, 2) \][/tex]
Thus,
[tex]\[ \int_C \vec{F} \cdot d\vec{r} = 4 - 6 = -2 \][/tex]
Final Answer:
[tex]\[ \boxed{-2} \][/tex]
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. For clear and precise answers, choose IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.
