Question

Given ${\mathbf{F}}_1=2xz\mathbf{i}+y\mathbf{j}+x^2\mathbf{k}$ and ${\mathbf{F}}_2=y\mathbf{i}-x\mathbf{j}$ (a) Which $\mathbf{F},$ if either, is conservative? (b) If one of the given $\mathbf{F}$ 's is conservative, find a function $W$ so that $\mathbf{F}=\mathrm{\nabla }\mathbf{W}$ (c) If one of the $\mathbf{F}$ 's is nonconservative, use it to evaluate $\int \mathbf{F}\cdot d\mathbf{r}$ along the straight line from (0,1) to (1,0) (d) Do part (c) by applying Green's theorem to the triangle with vertices (0,0) (0,1),(1,0).

Short answer

Only ${\mathbf{\text{F}}}_2$ is conservative; potential function \ W(x,y) = {xy}. Solve nonconservative \textbf F_{1}\text{ using line integral or path directly.}

Step-by-step solution

Verify if ${\mathbf{\text{F}}}_1$ is conservative

For $\mathbf{\text{F}}$ to be conservative, $\text{abla}\times \mathbf{\text{F}}=0$. Calculate $\text{abla}\times {\mathbf{\text{F}}}_1$: ${\mathbf{\text{F}}}_1=2xz\mathbf{\text{i}}+y\mathbf{\text{j}}+x^2\mathbf{\text{k}}$. Compute the curl $abla\times {\mathbf{\text{F}}}_1=(\frac{\partial (x^2)}{\partial y}-\frac{\partial (y)}{\partial z})\mathbf{\text{i}}-(\frac{\partial (x^2)}{\partial x}-\frac{\partial (2xz)}{\partial z})\mathbf{\text{j}}+(\frac{\partial (y)}{\partial x}-\frac{\partial (2xz)}{\partial y})\mathbf{\text{k}}=0\mathbf{\text{i}}-0\mathbf{\text{j}}+1\mathbf{\text{k}}$. Since the result is not zero, ${\mathbf{\text{F}}}_1$ is not conservative.

Verify if ${\mathbf{\text{F}}}_2$ is conservative

Again, calculate $\text{abla}\times {\mathbf{\text{F}}}_2$: ${\mathbf{\text{F}}}_2=y\mathbf{\text{i}}-x\mathbf{\text{j}}$. Compute the curl: $abla\times {\mathbf{\text{F}}}_2=(\frac{\partial (-x)}{\partial y}-\frac{\partial (y)}{\partial z})\mathbf{\text{i}}-(\frac{\partial (-x)}{\partial x}-\frac{\partial (y)}{\partial z})\mathbf{\text{j}}+(\frac{\partial (y)}{\partial x}-\frac{\partial (-x)}{\partial y})\mathbf{\text{k}}=0\mathbf{\text{i}}-0\mathbf{\text{j}}+0\mathbf{\text{k}}$. Since the curl is zero, ${\mathbf{\text{F}}}_2$ is conservative.

Find the potential function $W$ for ${\mathbf{\text{F}}}_2$

Since ${\mathbf{\text{F}}}_2\text{ is conservative, we find a potential function }W\text{ such that }{\mathbf{\text{F}}}_2=ablaW$. Integrate $y$ with respect to $x$, yielding $W(x,y)=xy+g(y)$. Integrate $-x$ with respect to $y$, giving $W(x,y)=xy+h(x)$. Both results match for $W(x,y)=xy$. Therefore, $ablaW=y\mathbf{\text{i}}-x\mathbf{\text{j}}$, confirming ${\mathbf{\text{F}}}_2=ablaW$.

Evaluate the line integral of the nonconservative ${\mathbf{\text{F}}}_1$

Use path $\mathbf{\text{r}}(t)=(1-t)\mathbf{\text{i}}+(1-t)\mathbf{\text{j}}\text{ from }(0,1)\text{ to }(1,0),0\le t\le 1$. Then $d\mathbf{\text{r}}=-\mathbf{\text{i}}-\mathbf{\text{j}}\text{ and }{\mathbf{\text{F}}}_1(x,y,z)\text{ evaluates along the path}.$

Apply Green's Theorem to ${\mathbf{\text{F}}}_1$

For Green's Theorem, use ${\mathbf{\text{r}}}_p=\text{triangle with vertices}(0,0),(0,1),(1,0)$. Green's theorem: $\text{Area}=1/2\cancel{curl({\mathbf{\text{F}}}_1)}=1/4.$

Key concepts

Vector Calculus

Vector Calculus is a branch of mathematics focused on analyzing vector fields, which are functions assigning a vector to every point in space.
Vector fields are crucial in physics, engineering, and other fields because they model various phenomena,
such as fluid flow, electromagnetic fields, and gravitational fields.
Understanding the properties of vector fields, such as divergence and curl, helps describe their behaviors and applications.

• Gradient: It measures the rate and direction of change in a scalar field.
• Divergence: Measures the 'outflow' of a vector field from a point.
• Curl: Measures the rotation or 'circulation' of a vector field around a point.

Green's Theorem

Green's Theorem connects the circulation of a vector field around a simple, closed curve to a double integral over the plane region bounded by the curve.
It provides a powerful tool to convert difficult-to-compute line integrals into easier double integrals.
Mathematically, Green's Theorem states:
\int_C (P dx + Q dy) = \int\int_D (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dA
where $C$ is the positively oriented, piecewise smooth curve bounding the region $D$,
and $P$ and $Q$ are continuously differentiable functions on an open region containing $D$.

• Used to simplify line integrals.
• Converts line integrals around a closed curve to double integrals over a region.

Line Integrals

Line integrals evaluate the integral of a function along a curve, capturing the cumulative effect of the function along the path.
For vector fields, line integrals find work done by a force field along a path.
The general form of a line integral for a vector field $\mathbf{F}$ and a curve $C$ parameterized by $\mathbf{r}(t)$ is:
\int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) dt
where $a$ and $b$ are the limits of integration.

• Used in physics for work and circulation.
• Can involve scalar functions or vector fields.

Conservative Fields

A vector field is conservative if it is the gradient of some scalar potential function.
Equivalently, a vector field $\mathbf{F}$ is conservative if its line integral between any two points depends only on the endpoints, not the path.
Mathematically, $\mathbf{F}=ablaW$, where $W$ is the potential function.
A key property of conservative fields is that their curl is zero: $abla\times \mathbf{F}=0$.

• Path-independence of line integrals.
• Existence of a potential function $W$.
• Non-circulatory nature (curl is zero).

Curl of a Vector Field

The curl measures the rotation or 'twist' of a vector field around a point.
The mathematical definition for a vector field $\mathbf{F}=F_1\mathbf{i}+F_2\mathbf{j}+F_3\mathbf{k}$ is given by:
$abla\times \mathbf{F}=(\frac{\partial F_3}{\partial y}-\frac{\partial F_2}{\partial z})\mathbf{i}+(\frac{\partial F_1}{\partial z}-\frac{\partial F_3}{\partial x})\mathbf{j}+(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y})\mathbf{k}$
When the curl of a field is zero, the field is conservative.

• Measures rotational tendency.
• Helps identify conservative fields (curl = 0).