Question

1. Let $\mathbf{F}=(ye^x+\sin y)\mathbf{i}+(e^x+x\cos y)\mathbf{j}$. (a) Verify that $\mathbf{F}$ is conservative. (b) Determine a potential $\varphi$ for $\mathbf{F}$. 2. Let $\mathbf{F}=(2x+y,x)$. (a) Verify that $\mathbf{F}$ is conservative, and find a potential $\varphi$ for $\mathbf{F}$. (b) Calculate $\int_C \mathbf{F}\cdot d\mathbf{r}$, where $C$ is the straight line path from $(1,0)$ to $(0,-1)$. (c) Calculate $\int_C \mathbf{F}\cdot d\mathbf{r}$, where $C$ is the path from $(1,0)$ to $(0,-1)$ counterclockwise along the circle $x^2+y^2=1$. (d) Calculate $\varphi(0,-1)-\varphi(1,0)$, and compare it to your answers from parts (b) and (c). What do you notice? 3. Let $\mathbf{F}=(yz,xz,xy+2z)$. (a) Verify that $\mathbf{F}$ is conservative. (b) Verify that $\varphi(x,y,z)=xyz+z^2$ is a potential for $\mathbf{F}$. (c) Evaluate the line integral $\int_C \mathbf{F}\cdot d\mathbf{r}$ where $C$ is the line segment from $(1,0,1)$ to $(0,-1,2)$. (d) Calculate $\varphi(0,-1,2)-\varphi(1,0,1)$, and compare this to your answer to part (c). 4. Consider the vector field $\mathbf{F}(x,y,z)=y^2\mathbf{i}+(2xy+e^z)\mathbf{j}+3ye^z\mathbf{k}$. (a) Demonstrate that $\mathbf{F}$ is conservative. (b) Find a potential $\varphi$ for $\mathbf{F}$.

          1. Let $\mathbf{F}=(ye^x+\sin y)\mathbf{i}+(e^x+x\cos y)\mathbf{j}$.
(a) Verify that $\mathbf{F}$ is conservative.
(b) Determine a potential $\varphi$ for $\mathbf{F}$.

2. Let $\mathbf{F}=(2x+y,x)$.
(a) Verify that $\mathbf{F}$ is conservative, and find a potential $\varphi$ for $\mathbf{F}$.
(b) Calculate $\int_C \mathbf{F}\cdot d\mathbf{r}$, where $C$ is the straight line path from $(1,0)$ to $(0,-1)$.
(c) Calculate $\int_C \mathbf{F}\cdot d\mathbf{r}$, where $C$ is the path from $(1,0)$ to $(0,-1)$ counterclockwise along the circle $x^2+y^2=1$.
(d) Calculate $\varphi(0,-1)-\varphi(1,0)$, and compare it to your answers from parts (b) and (c). What do you notice?

3. Let $\mathbf{F}=(yz,xz,xy+2z)$.
(a) Verify that $\mathbf{F}$ is conservative.
(b) Verify that $\varphi(x,y,z)=xyz+z^2$ is a potential for $\mathbf{F}$.
(c) Evaluate the line integral $\int_C \mathbf{F}\cdot d\mathbf{r}$ where $C$ is the line segment from $(1,0,1)$ to $(0,-1,2)$.
(d) Calculate $\varphi(0,-1,2)-\varphi(1,0,1)$, and compare this to your answer to part (c).

4. Consider the vector field
$\mathbf{F}(x,y,z)=y^2\mathbf{i}+(2xy+e^z)\mathbf{j}+3ye^z\mathbf{k}$.
(a) Demonstrate that $\mathbf{F}$ is conservative.
(b) Find a potential $\varphi$ for $\mathbf{F}$.

1 let mathbffyexsin ymathbfiexxcos ymathbfj a verify that mathbff is conservative b determine a potential varphi for mathbff 2 let mathbff2xyx a verify that mathbff is conservative and find 48385

Added by Laura O.

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