Given Show that
Given
$$ \vec{F}(x, y)=\left\langle\frac{2 x^{3}+2 x y^{2}-2 y}{x^{2}+y^{2}}, \frac{2 y^{3}+2 x^{2} y+2 x}{x^{2}+y^{2}}\right\rangle $$
Show that \(\int_{C} \vec{F} \cdot d \vec{r}=4 \pi\) for any positively oriented simple closed curve that encloses the origin.
Homework Answers
Find for the given F and C
Find \(\int_{C} \vec{F} \cdot d r\) for the given \(\vec{F}\) and \(C\).\(\cdot \vec{F}=-y \vec{i}+x \vec{j}+7 \vec{k}\) and \(C\) is the helix \(x=\cos t, y=\sin t r \quad z=t\), for \(0 \leq t \leq 2 \pi .\)$$ \int_{C} \vec{F} \cdot d \vec{r}= $$Find \(\int_{C} \overrightarrow{\mathrm{F}} \cdot d \overrightarrow{\mathrm{r}}\) for \(\overrightarrow{\mathrm{F}}=e^{y} \overrightarrow{\mathrm{i}}+\ln \left(x^{2}+1\right) \overrightarrow{\mathrm{j}}+\overrightarrow{\mathrm{k}}\) and \(C\), the circle of radius 4 centered at the origin in the \(y z\)-plane as shown below.$$ \int_{C} \vec{F} \cdot d \vec{r}= $$
Exact answer/fraction
Evaluate \(\int_{C} \mathrm{~F} \cdot d \mathbf{r}\) using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results.$$ \int_{C}[4(2 x+7 y) \mathbf{i}+14(2 x+7 y) \mathbf{j}] \cdot d \mathbf{r} $$C: smooth curve from \((-7,2)\) to \((3,2)\)Evaluate \(\int_{C} \mathrm{~F} \cdot d \mathbf{r}\) using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results.$$ \int_{C} \cos (x) \sin (y) d x+\sin (x) \cos (y) d y $$C: line segment from \((0,-\pi)\) to \(\left(\frac{3 \pi}{2},...
Consider the region
4. [5 pts.] Consider the region \(D\), outside the circles \(C_{2}\) and \(C_{3}\) and inside the circle \(C_{1}\) in the figure below and a vector field \(\vec{F}(x, y)=\langle P(x, y), Q(x, y)\rangle\). Assume we know that \(\oint_{C_{2}} \vec{F} \cdot d \vec{r}=\oint_{C_{3}} \vec{F} \cdot d \vec{r}=-2 \pi\), and \(Q_{x}-P_{y}=2\) on an open region containing \(D .\) UseGreen's Theorem to find \(\oint_{C_{1}} \vec{F} \cdot d \vec{r}\).
Use Stokes' Theorem to make the following circulation calculations.
Suppose \(\vec{F}=(5 x-3 y) \vec{i}+(x+4 y) \vec{j}\). Use Stokes' Theorem to make the following circulation calculations.(a) Find the circulation of \(\vec{F}\) around the circle \(C\) of radius 10 centered at the origin in the xy-plane, oriented clockwise as viewed from the positive z-axis. Circulation \(=\int_{C} \vec{F} \cdot d \vec{r}=\)(b) Find the circulation of \(\vec{F}\) around the circle \(C\) of radius 10 centered at the origin in the yz-plane, oriented clockwise as viewed from the positive \(x\)-axis. Circulation \(=\int_{C} \vec{F} \cdot...
Consider a solid of N atoms in contact with ta heat reservoir in the temperature region where the Debye T3 law is valid.
4. Consider a solid of \(N\) atoms in contact with ta heat reservoir in the temperature region where the Debye \(T^{3}\) law is valid. Show that the energy fluctuation is given by$$ \frac{\left\langle E^{2}\right\rangle-\langle E\rangle^{2}}{\langle E\rangle^{2}} \approx \frac{0.07}{N}\left(\frac{\theta_{D}}{T}\right)^{3} $$
Evaluate line integral ( F. dr where C is any positively oriented simple closed curve that...
Evaluate line integral ( F. dr where C is any positively oriented simple closed curve that encloses the origin by using a circle of radius r, and r is small enough so that the circle lies entirely inside C given F(x, y) = ? 1)_ 2xyi +(y2 – xº)j Ans (x² + y²)
Show that a helix of radius R and height h making N complete turns has the parametrization
Let\(\mathbf{r}(t)=\left\langle R \cos \left(\frac{2 \pi N t}{h}\right), R \sin \left(\frac{2 \pi N t}{h}\right), t\right\rangle, \quad 0 \leq t \leq h\)(a) Show that \(\mathbf{r}(t)\) parametrizes a helix of radius \(R\) and height \(h\) making \(N\) complete turns.(b) Guess which of the two springs in Figure 5 uses more wire.(c) Compute the lengths of the two springs and compare.
value of z= 96 Task 3: Answer the following
value of z= 96Task 3: Answer the following:a. Evaluate: \(\int_{\frac{\pi}{2}}^{\pi} \boldsymbol{Z} \cos ^{3}(x) \sin ^{2}(x) d x\)b. The moment of inertia, \(I\), of \(a\) rod of mass ' \(m^{\prime}\) and length \(4 r\) is given by \(I=\int_{0}^{4 r}\left(\frac{Z m x^{2}}{2 r}\right) d x\) where \(^{\prime} x^{\prime}\) is the distance from an axis of rotation. Find \(I \)Task 4: Answer the following:Using the Trapezoidal rule, find the approximate the area bounded by the curve\(y=\boldsymbol{Z} e^{\left(\frac{x}{2}\right)}\), the \(\mathrm{x}\) -axis and coordinates \(x=0,...
Consider the vector field: f (x, y)= «M(x, y), N(x, y)= v promet Let C be...
Consider the vector field: f (x, y)= «M(x, y), N(x, y)= v promet Let C be any simple, positively oriented, closed curve that encloses the origin. Show that: F. do 21. We will solve this problem by completing the following steps: STEP 1 Let C be a positively oriented circle of radius r with the center at the origin. Letr be so small that the circle Člies within the region enclosed by the curve C(see figure below) Compute the integral...
15. Let F(z,y)- F dr where C is any positively-oriented Jordan curve that encloses the origin Evaluate 15. Let F(z,y)- F dr where C is any positively-oriented Jordan curve that encloses the o...
15. Let F(z,y)- F dr where C is any positively-oriented Jordan curve that encloses the origin Evaluate
15. Let F(z,y)- F dr where C is any positively-oriented Jordan curve that encloses the origin Evaluate
4. [5 pts.] Consider the region \(D\), outside the circles \(C_{2}\) and \(C_{3}\) and inside the circle \(C_{1}\) in the figure below and a vector field \(\vec{F}(x, y)=\langle P(x, y), Q(x, y)\rangle\). Assume we know that \(\oint_{C_{2}} \vec{F} \cdot d \vec{r}=\oint_{C_{3}} \vec{F} \cdot d \vec{r}=-2 \pi\), and \(Q_{x}-P_{y}=2\) on an open region containing \(D .\) UseGreen's Theorem to find \(\oint_{C_{1}} \vec{F} \cdot d \vec{r}\).
Evaluate line integral ( F. dr where C is any positively oriented simple closed curve that encloses the origin by using a circle of radius r, and r is small enough so that the circle lies entirely inside C given F(x, y) = ? 1)_ 2xyi +(y2 – xº)j Ans (x² + y²)
value of z= 96Task 3: Answer the following:a. Evaluate: \(\int_{\frac{\pi}{2}}^{\pi} \boldsymbol{Z} \cos ^{3}(x) \sin ^{2}(x) d x\)b. The moment of inertia, \(I\), of \(a\) rod of mass ' \(m^{\prime}\) and length \(4 r\) is given by \(I=\int_{0}^{4 r}\left(\frac{Z m x^{2}}{2 r}\right) d x\) where \(^{\prime} x^{\prime}\) is the distance from an axis of rotation. Find \(I \)Task 4: Answer the following:Using the Trapezoidal rule, find the approximate the area bounded by the curve\(y=\boldsymbol{Z} e^{\left(\frac{x}{2}\right)}\), the \(\mathrm{x}\) -axis and coordinates \(x=0,...
Consider the vector field: f (x, y)= «M(x, y), N(x, y)= v promet Let C be any simple, positively oriented, closed curve that encloses the origin. Show that: F. do 21. We will solve this problem by completing the following steps: STEP 1 Let C be a positively oriented circle of radius r with the center at the origin. Letr be so small that the circle Člies within the region enclosed by the curve C(see figure below) Compute the integral...
15. Let F(z,y)- F dr where C is any positively-oriented Jordan curve that encloses the origin Evaluate
15. Let F(z,y)- F dr where C is any positively-oriented Jordan curve that encloses the origin Evaluate
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