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study help
mathematics
calculus 4th
Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions
Compute \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) for the oriented curve specified.\(\mathbf{F}(x, y)=\left\langle 1+x^{2}, x y^{2}ightangle\), line segment from \((0,0)\) to \((1,3)\)
Compute \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) for the oriented curve specified.\(\mathbf{F}(x, y)=\langle-2, yangle\), half-circle \(x^{2}+y^{2}=1\) with \(y \geq 0\), oriented counterclockwise
Compute \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) for the oriented curve specified.\(\mathbf{F}(x, y)=\left\langle x^{2}, x yightangle\), part of circle \(x^{2}+y^{2}=9\) with \(x \leq 0, y \geq 0\), oriented clockwise
Compute \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) for the oriented curve specified.\(\mathbf{F}(x, y)=\left\langle e^{y-x}, e^{2 x}ightangle\), piecewise linear path from \((1,1)\) to \((2,2)\) to \((0,2)\)
Compute \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) for the oriented curve specified.\(\mathbf{F}(x, y)=\left\langle 3 z y^{-1}, 4 x,-yightangle, \quad \mathbf{r}(t)=\left\langle e^{t}, e^{t}, tightangle\) for \(-1 \leq t \leq 1\)
Compute \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) for the oriented curve specified.\(\mathbf{F}(x, y)=\left\langle\frac{-y}{\left(x^{2}+y^{2}ight)^{2}}, \frac{x}{\left(x^{2}+y^{2}ight)^{2}}ightangle\), circle of radius \(R\) with center at the origin oriented counterclockwise
Compute \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) for the oriented curve specified.\(\mathbf{F}(x, y, z)=\left\langle\frac{1}{y^{3}+1}, \frac{1}{z+1}, 1ightangle, \quad \mathbf{r}(t)=\left\langle t^{3}, 2, t^{2}ightangle\) for \(0 \leq t \leq 1\)
Compute \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) for the oriented curve specified.\(\mathbf{F}(x, y, z)=\left\langle z^{3}, y z, xightangle\), quarter of the circle of radius 2 in the \(y z\)-plane with center at the origin where \(y \geq 0\) and \(z \geq 0\), oriented clockwise when viewed from
Evaluate the line integral.\(\int_{C} x d x\), over \(y=x^{3}\) for \(0 \leq x \leq 3\)
Evaluate the line integral.\(\int_{C} y d y\), over \(y=x^{3}\) for \(0 \leq x \leq 3\)
Evaluate the line integral.\(\int_{C} y d x-x d y\), parabola \(y=x^{2}\) for \(0 \leq x \leq 2\)
Evaluate the line integral.\(\int_{C} y d x+z d y+x d z, \quad \mathbf{r}(t)=\left\langle 2+t^{-1}, t^{3}, t^{2}ightangle\) for \(0 \leq t \leq 1\)
Evaluate the line integral.\(\int_{C}(x-y) d x+(y-z) d y+z d z, \quad\) line segment from \((0,0,0)\) to \((1,4,4)\)
Evaluate the line integral.\(\int_{C} z d x+x^{2} d y+y d z, \quad \mathbf{r}(t)=\langle\cos t, \tan t, tangle\) for \(0 \leq t \leq \frac{\pi}{4}\)
Evaluate the line integral.\(\int_{C} \frac{-y d x+x d y}{x^{2}+y^{2}}\), segment from \((1,0)\) to \((0,1)\)
Evaluate the line integral.\(\int_{C} y^{2} d x+z^{2} d y+\left(1-x^{2}ight) d z\), quarter of the circle of radius 1 in the \(x z\)-plane with center at the origin in the quadrant \(x \geq 0, z \leq 0\), oriented counterclockwise when viewed from the positive \(y\)-axis
CBS Let \(f(x, y, z)=x^{-1} y z\), and let \(C\) be the curve parametrized by \(\mathbf{r}(t)=\left\langle\ln t, t, t^{2}ightangle\) for \(2 \leq t \leq 4\). Use a computer algebra system to calculate \(\int_{C} f(x, y, z) d s\) to four decimal places.
185 Use a CAS to calculate \(\int_{C}\left\langle e^{x-y}, e^{x+y}ightangle \cdot d \mathbf{r}\) to four decimal places, where \(C\) is the curve \(y=\sin x\) for \(0 \leq x \leq \pi\), oriented from left to right.
Calculate the line integral of \(\mathbf{F}(x, y, z)=\left\langle e^{z}, e^{x-y}, e^{y}ightangle\) over the given path.The blue path from \(P\) to \(Q\) in Figure 15 (0,0,1). P = (0,0,0) (0, 1,1) Q=(-1,1,1)
Calculate the line integral of \(\mathbf{F}(x, y, z)=\left\langle e^{z}, e^{x-y}, e^{y}ightangle\) over the given path.The closed path \(A B C A\) in Figure 16 A = (2, 0, 0), C = (0, 0, 6) B = (0,4, 0) y
\(C\) is the path from \(P\) to \(Q\) in Figure 17 that traces \(C_{1}, C_{2}\), and \(C_{3}\) in the orientation indicated, and \(\mathbf{F}\) is a vector field such that\[\int_{C} \mathbf{F} \cdot d \mathbf{r}=5, \quad \int_{C_{1}} \mathbf{F} \cdot d \mathbf{r}=8, \quad \int_{C_{3}} \mathbf{F}
\(C\) is the path from \(P\) to \(Q\) in Figure 17 that traces \(C_{1}, C_{2}\), and \(C_{3}\) in the orientation indicated, and \(\mathbf{F}\) is a vector field such that\[\int_{C} \mathbf{F} \cdot d \mathbf{r}=5, \quad \int_{C_{1}} \mathbf{F} \cdot d \mathbf{r}=8, \quad \int_{C_{3}} \mathbf{F}
The values of a function \(f(x, y, z)\) and vector field \(\mathbf{F}(x, y, z)\) are given at six sample points along the path \(A B C\) in Figure 18. Estimate the line integrals of \(f\) and \(\mathbf{F}\) along \(A B C\). Point (1,2,0) (1, 1,0) (1,3,0) (1,1,1) (1, 1, 1) (1,1,2) f(x,y,z) 3 3.3 3.6
Estimate the line integrals of \(f(x, y)\) and \(\mathbf{F}(x, y)\) along the quarter-circle (oriented counterclockwise) in Figure 19 using the values at the three sample points along each path. Pointf(x,y) A B C y C 1 -2 4 B F(x, y) (1,2) (1,3) (-2,4) A 12
Determine whether the line integrals of the vector fields around the circle (oriented counterclockwise) in Figure 20 are positive, negative, or zero. 14 " A 2 + 1 1 1 * n 1 1 1 t 1 1 1 - " 1 1 T 1 1 f 1 f 1 1 + t + + 1 (B) (A) " F t (C)
Determine whether the line integrals of the vector fields along the oriented curves in Figure 21 are positive or negative. (A) (B) 1 7/11 (C)
Calculate the total mass of a circular piece of wire of radius \(4 \mathrm{~cm}\) centered at the origin whose mass density is \(ho(x, y)=x^{2} \mathrm{~g} / \mathrm{cm}\).
Calculate the total mass of a metal tube in the helical shape \(\mathbf{r}(t)=\left(\cos t, \sin t, t^{2}ight)\) (distance in centimeters) for \(0 \leq t \leq 2 \pi\) if the mass density is \(ho(x, y, z)=\sqrt{z} \mathrm{~g} / \mathrm{cm}\).
Find the total charge on the curve \(y=x^{4 / 3}\) for \(1 \leq x \leq 8\) (in centimeters) assuming a charge density of \(ho(x, y)=x / y\) (in units of \(10^{-6} \mathrm{C} / \mathrm{cm}\) ).
Find the total charge on the curve \(\mathbf{r}(t)=\left(\sin t, \cos t, \sin ^{2} tight)\) in centimeters for \(0 \leq t \leq \frac{\pi}{8}\) assuming a charge density of \(ho(x, y, z)=x y\left(y^{2}-zight)\) (in units of \(10^{-6} \mathrm{C} / \mathrm{cm}\) ).
Use Eq. (6) to compute the electric potential \(V(P)\) at the point \(P\) for the given charge density (in units of \(10^{-6} C\) ).Calculate \(V(P)\) at \(P=(0,0,12)\) if the electric charge is distributed along the quarter circle of radius 4 centered at the origin with charge density \(ho(x, y,
Use Eq. (6) to compute the electric potential \(V(P)\) at the point \(P\) for the given charge density (in units of \(10^{-6} C\) ).Calculate \(V(P)\) at the origin \(P=(0,0)\) if the negative charge is distributed along \(y=x^{2}\) for \(1 \leq x \leq 2\) with charge density \(ho(x, y)=-y
Use Eq. (6) to compute the electric potential V(P)V(P) at the point PP for the given charge density (in units of 10−6C10−6C ).Calculate \(V(P)\) at \(P=(2,0,2)\) if the negative charge is distributed along the \(y\)-axis for \(1 \leq y \leq 3\) with charge density \(ho(x, y, z)=-y\). V(P) = k
Use Eq. (6) to compute the electric potential V(P)V(P) at the point PP for the given charge density (in units of 10−6C10−6C ).Calculate \(V(P)\) at the origin \(P=(0,0)\) if the electric charge is distributed along \(y=x^{-1}\) for \(\frac{1}{2} \leq x \leq 2\) with charge density \(ho(x,
Calculate the work done by a field \(\mathbf{F}=\langle x+y, x-yangle\) when an object moves from \((0,0)\) to \((1,1)\) along each of the paths \(y=x^{2}\) and \(x=y^{2}\).
Calculate the work done by the field \(\mathbf{F}\) when the object moves along the given path from the initial point to the final point.\(\mathbf{F}(x, y, z)=\langle x, y, zangle, \mathbf{r}=\langle\cos t, \sin t, tangle\) for \(0 \leq t \leq 3 \pi\)
Calculate the work done by the field \(\mathbf{F}\) when the object moves along the given path from the initial point to the final point.\(\mathbf{F}(x, y, z)=\langle x y, y z, x zangle, \mathbf{r}=\left\langle t, t^{2}, t^{3}ightangle\) for \(0 \leq t \leq 1\)
Calculate the work done by the field \(\mathbf{F}\) when the object moves along the given path from the initial point to the final point.\(\mathbf{F}(x, y, z)=\left\langle e^{x}, e^{y}, x y zightangle, \mathbf{r}=\left\langle t^{2}, t, t / 2ightangle\) for \(0 \leq t \leq 1\)
Figure 22 shows a force field \(\mathbf{F}\).(a) Over which of the two paths, \(A D C\) or \(A B C\), does \(\mathbf{F}\) perform less work?(b) If you have to work against \(\mathbf{F}\) to move an object from \(C\) to \(A\), which of the paths, \(C B A\) or \(C D A\), requires less work? B A D
Verify that the work performed along the segment \(\overline{P Q}\) by the constant vector field \(\mathbf{F}=\langle 2,-1,4angle\) is equal to \(\mathbf{F} \cdot \overrightarrow{P Q}\) in these cases:(a) \(P=(0,0,0), Q=(4,3,5)\)(b) \(P=(3,2,3), Q=(4,8,12)\)
Show that work performed by a constant force field \(\mathbf{F}\) over any path \(C\) from \(P\) to \(Q\) is equal to \(\mathbf{F} \cdot \overrightarrow{P Q}\).
A curve \(C\) in polar form \(r=f(\theta)\) is parametrized by \(\mathbf{r}(\theta)=(f(\theta) \cos \theta, f(\theta) \sin \theta)\) ) because the \(x\) - and \(y\)-coordinates are given by \(x=r \cos \theta\) and \(y=r \sin \theta\).(a) Show that
Charge is distributed along the spiral with polar equation \(r=\theta\) for \(0 \leq \theta \leq 2 \pi\). The charge density is \(ho(r, \theta)=r\) (assume distance is in centimeters and charge in units of \(10^{-6} \mathrm{C} / \mathrm{cm}\) ). Use the result of Exercise 60 (a) to compute the
Let \(\mathbf{F}\) be the vortex field (Figure 24):Let \(a>0, b -) =(++2 ) 1x + y x + y F(x, y) =
Let \(\mathbf{F}\) be the vortex field (Figure 24):Let \(C\) be a curve in polar form \(r=f(\theta)\) for \(\theta_{1} \leq \theta \leq \theta_{2}\) [Figure 25(B)], parametrized by \(\mathbf{r}(\theta)=\) \((f(\theta) \cos \theta, f(\theta) \sin \theta))\) as in Exercise 60.(a) Show that the vortex
Use Eq. (9) to calculate the flux of the vector field across the curve specified.\(\mathbf{F}(x, y)=\langle-y, xangle\); upper half of the unit circle, oriented clockwise 1 ds=f"F F(r()) LCF (F-n) ds N(1) ||r'(1)|| |r' (1)|| dt = ["* F(r(1)) - N(C) de
Use Eq. (9) to calculate the flux of the vector field across the curve specified.\(\mathbf{F}(x, y)=\left\langle x^{2}, y^{2}ightangle ; \quad\) segment from \((3,0)\) to \((0,3)\), oriented upward (F-n) ds = ["1 F(r(1)). N(1) ||r'(z)|| |r' (1)|| dt = ["r F(r(1)) N(1) di
The product of 2 × 2 matrices A and B is the matrix AB defined byThe (i, j)-entry of A is the dot product of the ith row of A and the jth column of B. Prove that det(AB) = det(A) det(B). a b b' ( 5 ) ( 25 ) = C d d' A B aa' + bc' ca' + de' ab' + bd' cb' + dd' AB
Let Φ1 : D1 → D2 and Φ2 : D2 → D3 be C1 maps, and let Φ2 ◦ Φ1 : D1 → D3 be the composite map. Use the Multivariable Chain Rule and Exercise 49 to show thatData From Exercise 49The product of 2 × 2 matrices A and B is the matrix AB defined byThe (i, j)-entry of A is the dot product
Use Exercise 50 to prove thatVerify that Jac(I) = 1, where I is the identity map I(u, v) = (u, v).Data From Exercise 50Let Φ1 : D1 → D2 and Φ2 : D2 → D3 be C1 maps, and let Φ2 ◦ Φ1 : D1 → D3 be the composite map. Use the Multivariable Chain Rule and Exercise 49 to show thatData From
Let (x̅, y̅) be the centroid of a domain D. For λ > 0, let λD be the dilation of D, defined byλD = {(λx, λy) : (x, y) ∈ D}Use the Change of Variables Formula to prove that the centroid of λD is (λx̅, λy̅).
Find the centroid of the solid bounded by the xy-plane, the cylinder x2 + y2 = R2, and the plane x/R + z/H = 1.
Calculate the coordinate yCM of the centroid of solid (B) in Figure 4 defined by x2 + y2 ≤ 1 and 0 ≤ z ≤ 1/2 y + 3/2. H R (A) (B)
Find the center of mass of the cylinder x2 + y2 = 1 for 0 ≤ z ≤ 1, assuming a mass density of δ(x, y, z) = z.
Find the center of mass of the sector of central angle 2θ0 (symmetric with respect to the y-axis) in Figure 5, assuming that the mass density is δ(x, y) = x2. y 1200 -
Find the center of mass of the part of the ball x2 + y2 + z2 = 1 in the first octant, assuming a mass density of δ(x, y, z) = x.
Find a constant C such that p(x, y) = (C(4x-y + 3) if 0x 2 and 0 y 3 0 otherwise is a probability distribution and calculate P(X 1; Y 2).
Calculate P(3X + 2Y ≥ 6) for the probability density in Exercise 52.
The lifetimes X and Y (in years) of two machine components have joint probability densityWhat is the probability that both components are still functioning after 2 years? p(x, y) = 6 (5-x-y) if 0 x 5-y and 0 y 5 125 otherwise
An insurance company issues two kinds of policies: A and B. Let X be the time until the next claim of type A is filed, and let Y be the time (in days) until the next claim of type B is filed. The random variables have joint probability densityp(x, y) = 12e−4x−3yFind the probability that X ≤ Y.
Compute the Jacobian of the map (r, s) (e cosh(s), e' sinh(s))
Find a linear mapping Φ(u, v) that maps the unit square to the parallelogram in the xy-plane spanned by the vectors 3, −1 and 1, 4 . Then use the Jacobian to find the area of the image of the rectangle R = [0, 4] × [0, 3] under Φ.
Use the map + D(u, v (u, v) = (u 2 v u 2 v) to compute (0, ). = ((x - y) sin(x + y)) dx dy, where R is the square with vertices (7,0), (27, 7), (7, 27), and
Let D be the shaded region in Figure 6, and let F be the map u = y + x, (a) Show that F maps D to a rectangle R in the uv-plane. (b) Apply Eq. (7) in Section 15.6 with P = (1,7) to estimate Area(D). 9 8 87 7 65 6 5 v=y=x P = (1,7) y=x +6 /y=x +5 -y=9-x y=8-x X
Calculate the integral of ƒ(x, y) = e3x−2y over the parallelogram in Figure 7. y (1, 3) D (6,4) (5, 1) -X
Sketch the region D bounded by the curves y = 2/x, y = 1/(2x), y = 2x, y = x/2 in the first quadrant. Let F be the map u = xy, v = y/x from the xy-plane to the uv-plane. (a) Find the image of D under F. (b) Let = F-. Show that Jac(): 1 2|v| (c) Apply the Change of Variables Formula to prove the
Which of the following is a unit vector field in the plane?(a) F = (y, x) (b) F = (c) F = y -2 {x +y X {x +y 2 y X x +y x2 + y2
Sketch an example of a nonconstant vector field in the plane in which each vector is parallel to (1, 1).
Show that the vector field F = −z, 0, x is orthogonal to the position vector −−→ OP at each point P. Give an example of another vector field with this property.
Show that ƒ(x, y, z) = xyz is a potential function for (yz, xz, xy) and give an example of a potential function other than ƒ.
Compute and sketch the vector assigned to the points P = (1, 2) and Q = (−1, −1) by the vector field F = (x2, x).
Compute and sketch the vector assigned to the points P = (1, 2) and Q = (−1, −1) by the vector field F = (−y, x).
Compute and sketch the vector assigned to the points P = (0, 1, 1) and Q = (2, 1, 0) by the vector field F = (xy, z2, x).
Compute the vector assigned to the points P = (1, 1, 0) and Q = (2, 1, 2) by the vector fields er, er/r, and er/r2.
Sketch the following planar vector fields by drawing the vectors attached to points with integer coordinates in the rectangle −3 ≤ x ≤ 3, −3 ≤ y ≤ 3. Instead of drawing the vectors with their true lengths, scale them if necessary to avoid overlap.F = (1, 0)
Sketch the following planar vector fields by drawing the vectors attached to points with integer coordinates in the rectangle −3 ≤ x ≤ 3, −3 ≤ y ≤ 3. Instead of drawing the vectors with their true lengths, scale them if necessary to avoid overlap.F = (1, 1)
Sketch the following planar vector fields by drawing the vectors attached to points with integer coordinates in the rectangle −3 ≤ x ≤ 3, −3 ≤ y ≤ 3. Instead of drawing the vectors with their true lengths, scale them if necessary to avoid overlap.F = xi
Sketch the following planar vector fields by drawing the vectors attached to points with integer coordinates in the rectangle −3 ≤ x ≤ 3, −3 ≤ y ≤ 3. Instead of drawing the vectors with their true lengths, scale them if necessary to avoid overlap.F = yi
Sketch the following planar vector fields by drawing the vectors attached to points with integer coordinates in the rectangle −3 ≤ x ≤ 3, −3 ≤ y ≤ 3. Instead of drawing the vectors with their true lengths, scale them if necessary to avoid overlap.F = (0, x)
Sketch the following planar vector fields by drawing the vectors attached to points with integer coordinates in the rectangle −3 ≤ x ≤ 3, −3 ≤ y ≤ 3. Instead of drawing the vectors with their true lengths, scale them if necessary to avoid overlap.F = x2i + yj
Sketch the following planar vector fields by drawing the vectors attached to points with integer coordinates in the rectangle −3 ≤ x ≤ 3, −3 ≤ y ≤ 3. Instead of drawing the vectors with their true lengths, scale them if necessary to avoid overlap. F = X y \x + y x + y
Sketch the following planar vector fields by drawing the vectors attached to points with integer coordinates in the rectangle −3 ≤ x ≤ 3, −3 ≤ y ≤ 3. Instead of drawing the vectors with their true lengths, scale them if necessary to avoid overlap. =(- F = -y x + y X x + y
Match each of the following planar vector fields with the corresponding plot in Figure 10.F = (2, x) 2- 0 -2 2 O -2. 1. t X ta 27 Y 0 (A) (C) tt N. ta X 2+ 0 -2- 2+ -2- - 1 * 0 (B) (D) ta X X
Match each of the following planar vector fields with the corresponding plot in Figure 10.F = (2x + 2, y) 2- 0 -2 2 O -2. 1. t X ta 27 Y 0 (A) (C) tt N. ta X 2+ 0 -2- 2+ -2- - 1 * 0 (B) (D) ta X X
Match each of the following planar vector fields with the corresponding plot in Figure 10.F = (y, cos x) 2- 0 -2 2 O -2. 1. t X ta 27 Y 0 (A) (C) tt N. ta X 2+ 0 -2- 2+ -2- - 1 * 0 (B) (D) ta X X
Match each of the following planar vector fields with the corresponding plot in Figure 10.F = (x + y, x − y) 2- 0 -2 2 O -2. 1. t X ta 27 Y 0 (A) (C) tt N. ta X 2+ 0 -2- 2+ -2- - 1 * 0 (B) (D) ta X X
Match each three-dimensional vector field with the corresponding plot in Figure 11.F = (1, 1, 1) ( (A) (B) (D)
Match each three-dimensional vector field with the corresponding plot in Figure 11.F = (x, 0, z) ( (A) (B) (D)
Match each three-dimensional vector field with the corresponding plot in Figure 11.F = (x, y, z) ( (A) (B) (D)
Match each three-dimensional vector field with the corresponding plot in Figure 11.F = er ( (A) (B) (D)
A river 200 meters wide is modeled by the region in the xy-plane given by −100 ≤ x ≤ 100. The velocity vector field on the surface of the river is given by F = (−0.05x, 20 − 0.0001x)2 in meters per second. Determine the coordinates of those points that have the maximum speed.
The velocity vectors in kilometers per hour for the wind speed of a tornado near the ground are given by the vector field Determine the coordinates of those points where the wind speed is the highest. F = -y X le(x+y-1) e(x+y-1)/
Calculate div(F) and curl(F).23. F =x, y, z
Calculate div(F) and curl(F).F = (y, z, x)
Calculate div(F) and curl(F).F = (x − 2zx2, z − xy, z2x2)
Calculate div(F) and curl(F).Sin(x + z)i − yexzk
Calculate div(F) and curl(F).F = (yz, xz, xy)
Calculate div(F) and curl(F). F = = () NIX
Calculate div(F) and curl(F). F (e, sin x, cos x) =
Calculate div(F) and curl(F). X y F= [x2 + y2' x2 + y2' 12.0)
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