Problem 19
Question
Show that the given integral is independent of the path. Evaluate. $$ \int_{(1,1,1)}^{(2,4,8)} y z d x+x z d y+x y d z $$
Step-by-Step Solution
Verified Answer
The integral is 63 and is independent of the path.
1Step 1: Identify the Vector Field
The given integral is a line integral of the vector field \( oldsymbol{F} = (yz, xz, xy) \), where components are determined from the integral expression \( yz \,dx + xz \,dy + xy \,dz \).
2Step 2: Check for Conservative Field
To determine if the vector field is conservative, calculate the curl, \( abla \times \boldsymbol{F} \). The curl of a vector field \( \boldsymbol{F} = (P, Q, R) \) is \( abla \times \boldsymbol{F} = (\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) \).
3Step 3: Compute Partial Derivatives for Curl
Calculate the partial derivatives: \( \frac{\partial R}{\partial y} = x \), \( \frac{\partial Q}{\partial z} = x \), \( \frac{\partial P}{\partial z} = y \), \( \frac{\partial R}{\partial x} = y \), \( \frac{\partial Q}{\partial x} = z \), \( \frac{\partial P}{\partial y} = z \).
4Step 4: Evaluate the Curl
Substitute the partial derivatives into the curl formula: \( (\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) = (x - x, y - y, z - z) = (0, 0, 0) \).
5Step 5: Conclude Path Independence
Since the curl is zero everywhere in the domain, \( abla \times \boldsymbol{F} = 0 \), the vector field is conservative, indicating the integral is path-independent.
6Step 6: Evaluate the Integral using Fundamental Theorem of Line Integrals
As \( \boldsymbol{F} \) is conservative, find a potential function \( f \) such that \( abla f = \boldsymbol{F} \). Assume \( f(x, y, z) = xyz \). Verify by computing \( abla f = (yz, xz, xy) \), which matches \( \boldsymbol{F} \). Evaluate \( f(2, 4, 8) - f(1, 1, 1) \).
7Step 7: Calculate the Potential Function Values
Compute \( f(2, 4, 8) = 2 \cdot 4 \cdot 8 = 64 \) and \( f(1, 1, 1) = 1 \cdot 1 \cdot 1 = 1 \).
8Step 8: Compute the Integral Result
Subtract the potential function values: \( 64 - 1 = 63 \).
Key Concepts
Vector FieldConservative FieldPotential FunctionFundamental Theorem of Line Integrals
Vector Field
A vector field is a function that assigns a vector to every point in space. In simpler terms, imagine a field of arrows, where each arrow has a direction and a magnitude, pointing at various locations in space. These arrows collectively form a vector field. It is a crucial concept in understanding how forces or velocities vary over a spatial region.
In the given problem, the vector field \( \boldsymbol{F} \) is represented by \( (yz, xz, xy) \). This representation helps us to visualize how each of the components—dependent on certain variables—influences the direction and magnitude of the vector field across its domain.
In the given problem, the vector field \( \boldsymbol{F} \) is represented by \( (yz, xz, xy) \). This representation helps us to visualize how each of the components—dependent on certain variables—influences the direction and magnitude of the vector field across its domain.
Conservative Field
A conservative field is one in which the line integral between two points does not depend on the path taken to connect them. This means that the net work done along any closed path in such a field is zero. The hallmark of a conservative field is that it has no "curl," or rotational component.
To determine if a field is conservative, we calculate its curl. For \( \boldsymbol{F} = (yz, xz, xy) \), the partial derivatives are \( abla \times \boldsymbol{F} = (0, 0, 0) \), meaning the curl is zero everywhere. Thus, \( \boldsymbol{F} \) is a conservative field, affirming that the line integral of \( \boldsymbol{F} \) is independent of the path.
To determine if a field is conservative, we calculate its curl. For \( \boldsymbol{F} = (yz, xz, xy) \), the partial derivatives are \( abla \times \boldsymbol{F} = (0, 0, 0) \), meaning the curl is zero everywhere. Thus, \( \boldsymbol{F} \) is a conservative field, affirming that the line integral of \( \boldsymbol{F} \) is independent of the path.
Potential Function
A potential function gives us a scalar value at every point in space and acts as an "overlay" from which a conservative vector field such as \( \boldsymbol{F} \) can be derived. Thus, if a vector field is conservative, there exists a potential function \( f \) such that \( abla f = \boldsymbol{F} \).
In our case, assume \( f(x, y, z) = xyz \). By computing the gradient \( abla f = (yz, xz, xy) \), we verify that it indeed matches \( \boldsymbol{F} \). Thus, \( f(x, y, z) = xyz \) is a valid potential function, enabling us to compute the line integral easily via this function.
In our case, assume \( f(x, y, z) = xyz \). By computing the gradient \( abla f = (yz, xz, xy) \), we verify that it indeed matches \( \boldsymbol{F} \). Thus, \( f(x, y, z) = xyz \) is a valid potential function, enabling us to compute the line integral easily via this function.
Fundamental Theorem of Line Integrals
The Fundamental Theorem of Line Integrals (FTLI) states that if \( \boldsymbol{F} \) is a conservative vector field with a potential function \( f \), then the line integral of \( \boldsymbol{F} \) from point A to point B is simply the difference \( f(B) - f(A) \).
This theorem simplifies the computation of a line integral, reducing it to evaluating a function at two points rather than integrating along a curve. In our solution, with \( f(x, y, z) = xyz \), we calculate \( f(2, 4, 8) = 64 \) and \( f(1, 1, 1) = 1 \). Hence, the integral result is \( 64 - 1 = 63 \), leveraging the simplicity the FTLI provides.
This theorem simplifies the computation of a line integral, reducing it to evaluating a function at two points rather than integrating along a curve. In our solution, with \( f(x, y, z) = xyz \), we calculate \( f(2, 4, 8) = 64 \) and \( f(1, 1, 1) = 1 \). Hence, the integral result is \( 64 - 1 = 63 \), leveraging the simplicity the FTLI provides.
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