1. ( 8 points) An object moves though a vector field, \(\overrightarrow{\mathbf{F}}(x, y)\), along a circular path, \(\overrightarrow{\mathbf{r}}(t)\), starting at \(P\) and ending at \(Q\) as shown in the graph below.
(a) At the point \(R\) draw and label a tangent vector in the direction of \(d \overrightarrow{\mathbf{r}}\).
(b) At the point \(R\) draw and label a vector in the direction of the vector filed, \(\overrightarrow{\mathbf{F}}(R)\).
(c) At the point \(R\) is \(\overrightarrow{\mathbf{F}} \cdot d \overrightarrow{\mathbf{r}}\) positive, negative, or zero? Circle the correct answer.
\(\begin{array}{lll}\text { NEGATIVE } & \text { ZERO } & \text { POSITIVE }\end{array}\)
(d) Is the line integral \(\int_{C} \overrightarrow{\mathbf{F}}(x, y) \cdot d \overrightarrow{\mathbf{r}}\) positive, negative or zero? Circle the correct answer.
\(\begin{array}{lll}\text { NEGATIVE } & \text { ZERO } & \text { POSITIVE }\end{array}\)
(e) Is the line integral \(\int_{-C} \overrightarrow{\mathbf{F}}(x, y) \cdot d \overrightarrow{\mathbf{r}}\) positive, negative or zero? Circle the correct answer. \(-C\) is the same path in the reverse direction.
NEGATIVE ZERO POSITIVE
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.$$ \int_{C} \vec{F} \cdot d \vec{r}= $$
I have the first method complete, but I can't figure out thesecond method. I have...
I have the first method complete, but I can't figure out the
second method Could
someone please show how to use the second method?2. Find the unit step response of:$$ \begin{aligned} \dot{\overrightarrow{\mathbf{x}}}(t) &=\left[\begin{array}{cc} 0 & 1 \\ -2 & -2 \end{array}\right] \overrightarrow{\mathbf{x}}(t)+\left[\begin{array}{l} 1 \\ 1 \end{array}\right] u(t) \\ y(t) &=\left[\begin{array}{cc} 2 & 3 \end{array}\right] \overrightarrow{\mathbf{x}}(t) \end{aligned} $$by two methods (1): transfer function and then (2) \(y(t)=\mathbf{C} e^{\mathbf{A} t} \overrightarrow{\mathbf{x}}(0)+\mathbf{C} \int_{0}^{t} e^{\mathbf{A}(t-\tau)} \mathbf{B} u(\tau) d \tau+\mathbf{D} u(t)\). Re-member that the Laplace...
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},...
Defining the cross product The cross product of two nonzero vectors
Defining the cross product The cross product of two nonzero vectors \(\vec{u}\) and \(\vec{v}\) is another vector \(\vec{u} \times \vec{v}\) with magnitude$$ |\vec{u} \times \vec{v}|=|\vec{u}||\vec{v}| \sin (\theta), $$where \(0 \leq \theta \leq \pi\) is the angle between the two vectors. The direction of \(\vec{u} \times \vec{v}\) is given by the right hand rule: when you put the vectors tail to tail and let the fingers of your right hand curl from \(\vec{u}\) to \(\vec{v}\) the direction of \(\vec{u} \times \vec{v}\)...
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...
Evaluate the surface integral
Evaluate the surface integral \(\iint_{S} F \cdot d \mathbf{S}\) for the given vector field \(\mathbf{F}\) and the oriented surface \(S\). In other words, find the flux of \(F\) across 5 . For closed surfaces, use the positive (outward) orientation.$$ \mathbf{F}(x, y, z)=x \mathbf{i}+3 y \mathbf{j}+2 z \mathbf{k} $$\(S\) is the cube with vertices \((\pm 1, \pm 1, \pm 1)\)
LE 4) (Ungraded) In Cartesian coordinates, the curl of a vector field Air) is defined as...
LE 4) (Ungraded) In Cartesian coordinates, the curl of a vector field Air) is defined as Use the definition of electric potential to find the potential difference between the origin and r = x + y + 27, V(r) - V(O) = - Ed. As the line integral is independent of path, choose whatever path you find to be con- vertient Taking V(0) = 0, what is V(r)? Finally, confirm that taking the gradient of the potential recovers our original...
point) Determine whether the line Integral of each vector field (n blue) along the semicircular, oriented...
point) Determine whether the line Integral of each vector field (n blue) along the semicircular, oriented path(in red is positive, negative of zero. LLLL Zero Negative : Positive Positive Positive (Click on a graph to enlarge it)
(1 point) Determine whether the line integral of each vector field (in blue) along the semicircular, oriented path (in red) is positive, negative, or zero Choose Choose v Choose (1 point) De...
(1 point) Determine whether the line integral of each vector field (in blue) along the semicircular, oriented path (in red) is positive, negative, or zero Choose Choose v Choose
(1 point) Determine whether the line integral of each vector field (in blue) along the semicircular, oriented path (in red) is positive, negative, or zero Choose Choose v Choose
(1 point) Consider the vector field F(x, y, z) = (2z + 3y)i + (2z +...
(1 point) Consider the vector field F(x, y, z) = (2z + 3y)i + (2z + 3x)j + (2y + 2x)k. a) Find a function f such that F = Vf and f(0,0,0) = 0. f(x, y, z) = b) Suppose C is any curve from (0,0,0) to (1,1,1). Use part a) to compute the line integral / F. dr. (1 point) Verify that F = V and evaluate the line integral of F over the given path: F =...
I have the first method complete, but I can't figure out the
second method Could
someone please show how to use the second method?2. Find the unit step response of:$$ \begin{aligned} \dot{\overrightarrow{\mathbf{x}}}(t) &=\left[\begin{array}{cc} 0 & 1 \\ -2 & -2 \end{array}\right] \overrightarrow{\mathbf{x}}(t)+\left[\begin{array}{l} 1 \\ 1 \end{array}\right] u(t) \\ y(t) &=\left[\begin{array}{cc} 2 & 3 \end{array}\right] \overrightarrow{\mathbf{x}}(t) \end{aligned} $$by two methods (1): transfer function and then (2) \(y(t)=\mathbf{C} e^{\mathbf{A} t} \overrightarrow{\mathbf{x}}(0)+\mathbf{C} \int_{0}^{t} e^{\mathbf{A}(t-\tau)} \mathbf{B} u(\tau) d \tau+\mathbf{D} u(t)\). Re-member that the Laplace...
Defining the cross product The cross product of two nonzero vectors \(\vec{u}\) and \(\vec{v}\) is another vector \(\vec{u} \times \vec{v}\) with magnitude$$ |\vec{u} \times \vec{v}|=|\vec{u}||\vec{v}| \sin (\theta), $$where \(0 \leq \theta \leq \pi\) is the angle between the two vectors. The direction of \(\vec{u} \times \vec{v}\) is given by the right hand rule: when you put the vectors tail to tail and let the fingers of your right hand curl from \(\vec{u}\) to \(\vec{v}\) the direction of \(\vec{u} \times \vec{v}\)...
LE 4) (Ungraded) In Cartesian coordinates, the curl of a vector field Air) is defined as Use the definition of electric potential to find the potential difference between the origin and r = x + y + 27, V(r) - V(O) = - Ed. As the line integral is independent of path, choose whatever path you find to be con- vertient Taking V(0) = 0, what is V(r)? Finally, confirm that taking the gradient of the potential recovers our original...
point) Determine whether the line Integral of each vector field (n blue) along the semicircular, oriented path(in red is positive, negative of zero. LLLL Zero Negative : Positive Positive Positive (Click on a graph to enlarge it)
(1 point) Determine whether the line integral of each vector field (in blue) along the semicircular, oriented path (in red) is positive, negative, or zero Choose Choose v Choose
(1 point) Determine whether the line integral of each vector field (in blue) along the semicircular, oriented path (in red) is positive, negative, or zero Choose Choose v Choose
(1 point) Consider the vector field F(x, y, z) = (2z + 3y)i + (2z + 3x)j + (2y + 2x)k. a) Find a function f such that F = Vf and f(0,0,0) = 0. f(x, y, z) = b) Suppose C is any curve from (0,0,0) to (1,1,1). Use part a) to compute the line integral / F. dr. (1 point) Verify that F = V and evaluate the line integral of F over the given path: F =...
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