The temperature at the point (x, y, z) in a substance with conductivity K = 6.5 is u (x, y, z) = 2y2 + 2z2. Find the rate of heat flow inward across the cylindrical surface y2 + z2 = 6, 0 < x < 4.
> (a). What does it mean to say that f is continuous at (a, b)? (b). If f is continuous on R2, what can you say about its graph?
> Find the gradient vector field of f. f (x, y, z) = x2yey/z
> Find the gradient vector field of f. f (x, y, z) = √ (x^2 + y^2 + z^2)
> Find the gradient vector field of f. f (s, t) = √ (2s + 3t)
> Find the gradient vector field of f. f (x, y) = y sin (xy)
> What is the connection between vector functions and space curves?
> If a is a constant vector, r = x i + y j + z k, and S is an oriented, smooth surface with a simple, closed, smooth, positively oriented boundary curve C, show that Sf 2a · ds = (a x r) · dr 2а
> Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. Figure 9: F (x, y) = y i + (x + y) j ZA
> Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. Figure 9: F (x, y, z) = i + k ZA
> Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. Figure 9: F (x, y) = (yi- xj)/√ (x^2+y^2) ZA
> Match the vector fields F on R3 with the plots labeled I–IV. Give reasons for your choices. F (x, y, z) = i + 2 j + z k II 1- -1 -1 0 y
> Match the vector fields F on R3 with the plots labeled I–IV. Give reasons for your choices. F (x, y, z) = i + 2 j + 3 k 1 z0- -1 -1 0 1 10-1 y
> Match the vector fields F with the plots labeled I–IV. Give reasons for your choices. F (x, y) = 〈cos (x + y), x〉 IV 3 -3 3 11 -3
> Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. Figure 9: F (x, y) = (yi + xj)/√ (x^2+y^2) ZA
> If z is defined implicitly as a function of x and y by an equation of the form F (x, y, z) = 0, how do you find ∂zy/∂x and ∂z/∂y?
> If z = f (x, y), what are the differentials dx, dy, and dz?
> Evaluate the line integral, where C is the given curve. ∫C x2y ds, C: x = cos t, y = sin t, z = t, 0 < t < π/2
> If the components of F have continuous second partial derivatives and S is the boundary surface of a simple solid region, show that ∫∫S curl F ∙ dS = 0.
> Investigate the shape of the surface with parametric equations x = sin u, y = sin v, z = sin (u + v). Start by graphing the surface from several points of view. Explain the appearance of the graphs by determining the traces in the horizontal planes z = 0
> Evaluate the line integral, where C is the given curve. ∫C x2 dx + y2 dy, C consists of the arc of the circle x2 + y2 = 4 from (2, 0) to (0, 2) followed by the line segment from (0, 2) to (4, 3)
> Evaluate the line integral, where C is the given curve. ∫C (x + 2y) dx + x2 dy, C consists of line segments from (0, 0) to (2, 1) and from (2, 1) to (3, 0)
> Evaluate the line integral, where C is the given curve. ∫C ex dx, C is the arc of the curve x = y3 from (-1, -1) to (1, 1)
> Evaluate the line integral, where C is the given curve. ∫C (x2y + sin x) dy, C is the arc of the parabola y = x2 from (0, 0) to (π, π2)
> Evaluate the line integral, where C is the given curve. ∫C xey ds, C is the line segment from (2, 0) to (5, 4)
> A thin wire is bent into the shape of a semicircle x2 + y2 = 4, x > 0. If the linear density is a constant k, find the mass and center of mass of the wire.
> Find the exact value of ∫C x3y2z ds, where C is the curve with parametric equations x = e-t cos 4t, y = e-t sin 4t, z = e-t, 0 < t < 2 π.
> (a). Evaluate the line integral∫C F ∙ dr, where F (x, y, z) = x i - z j + y k and C is given by r(t) = 2t i + 3t j - t2 k, -1 (b). Illustrate part (a) by using a computer to graph C and the vectors from the vector fie
> Evaluate the line integral, where C is the given curve. ∫C xy4 ds, C is the right half of the circle x2 + y2 = 16
> Use a calculator to evaluate the line integral correct to four decimal places. ∫C z ln (x + y) ds, where C has parametric equations x = 1 + 3t, y = 2 + t2, z = t4, -1 < t < 1
> Find ∫∫S F ∙ n dS, where F (x, y, z) = x i + y j + z k and S is the outwardly oriented surface shown in the figure (the boundary surface of a cube with a unit corner cube removed). ZA (0, 2, 2)
> Use a calculator to evaluate the line integral correct to four decimal places. ∫C xy arctan z ds, where C has parametric equations x = t2, y = t3, z = √t, 1 < t < 2
> (a). Find a parametric representation for the torus obtained by rotating about the z-axis the circle in the xz-plane with center (b, 0, 0) and radius a (b). Use the parametric equations found in part (a) to graph the torus for several values of a and b.
> Use a calculator to evaluate the line integral correct to four decimal places. ∫C F ∙ dr, where F (x, y) = √(x + y) i + (y/x) j and r(t) = sin2 t i + sin t cos t j, π/6 < t < π/3
> Evaluate the line integral ∫C F ∙ dr, where C is given by the vector function r(t). F (x, y, z) = x i + y j + xy k, r(t) = cos t i + sin t j + t k, 0 < t < π
> Evaluate the line integral ∫C F ∙ dr, where C is given by the vector function r(t). F (x, y, z) = sin x i + cos y j + xz k, r (t) = t3 i - t2 j + t k, 0 < t < 1
> (a). Show that the parametric equations x = a cosh u cos v, y = b cosh u sin v, z = c sinh u, represent a hyperboloid of one sheet. (b). Use the parametric equations in part (a) to graph the hyperboloid for the case a = 1, b = 2, c = 3. (c). Set up, but
> Evaluate the line integral, where C is the given curve. ∫C (x/y) ds, C: x = t 3, y = t 4, 1 < t < 2
> Evaluate the line integral ∫C F ∙ dr, where C is given by the vector function r(t). F (x, y) = xy2 i - x2 j, r(t) = t3 i + t2 j, 0 < t < 1
> Find the exact area of the surface z = 1 + 2x + 3y + 4y2, 1 < x < 4, 0 < y < 1.
> Find the area of the surface with vector equation r (u, v) = 〈cos3u cos3v, sin3u cos3v, sin3v〉, 0 < u < u, 0 < v < 2π. State your answer corrects to four decimal places.
> Let F (x, y) = (2x3 + 2xy2 - 2y) i + (2y3 + 2x2y + 2x) j x2 + y2 Evaluate ∫C F ∙ dr, where C is shown in the figure. C
> (a). Use the Midpoint Rule for double integrals (see Section 15.1) with six squares to estimate the area of the surface z = 1/ (1 + x2 + y2), 0 < x < 6, 0 < y < 4. (b). Use a computer algebra system to approximate the surface area in part (a) to four dec
> Find, to four decimal places, the area of the part of the surface z = (1 + x2)/ (1 + y2) that lies above the square |x | + |y | < 1. Illustrate by graphing this part of the surface.
> Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral. The part of the surface z = ln (x2 + y2 + 2) that lies above the disk x2 + y2 < 1
> Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral. The part of the surface z = cos (x2 + y2) that lies inside the cylinder x2 + y2 = 1
> If the equation of a surface S is z = f (x, y), where x2 + y2 < R2, and you know that | fx | < 1 and | fy | < 1, what can you say about A (S)?
> If C is a smooth curve given by a vector function r (t), a r• dr = {[Ir(b)|? – \r(a)F]
> Let F be an inverse square field, that is, F (r) = cr/|r |3 for some constant c, where r = x i + y j + z k. Show that the flux of F across a sphere S with center the origin is independent of the radius of S.
> The temperature at a point in a ball with conductivity K is inversely proportional to the distance from the center of the ball. Find the rate of heat flow across a sphere S of radius a with center at the center of the ball.
> Use Gauss’s Law to find the charge enclosed by the cube with vertices (±1, ±1, ±1) if the electric field is E (x, y, z) = x i + y j + z k
> Let F (x, y, z) = (3x2yz - 3y) i + (x3z - 3x) j + (x3y + 2z) k Evaluate ∫C F ∙ dr, where C is the curve with initial point (0, 0, 2) and terminal point (0, 3, 0) shown in the figure. ZA (0, 0, 2) (0, 3, 0) (1, 1,
> Use Gauss’s Law to find the charge contained in the solid hemisphere x2 + y2 + z2 < a2, z > 0, if the electric field is E (x, y, z) = x i + y j + 2z k
> Seawater has density 1025 kg/m3 and flows in a velocity field v = y i + x j, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward through the hemisphere x2 + y2 + z2 = 9, z > 0.
> A fluid has density 870 kg/m3 and flows with velocity v = z i + y2 j + x2 k, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward through the cylinder x2 + y2 = 4, 0 < z < 1.
> Find the area of the surface. The part of the cone z = √x2 + y2 that lies between the plane y = x and the cylinder y = x2
> (a). Give an integral expression for the moment of inertia Iz about the z-axis of a thin sheet in the shape of a surface S if the density function is ρ. (b). Find the moment of inertia about the z-axis of the funnel in Exercise 40 Exercise 40: Find the
> Find the mass of a thin funnel in the shape of a cone z = √x2 + y2, 1 < z < 4, if its density function is ρ (x, y, z) = 10 - z.
> Find the center of mass of the hemisphere x2 + y2 + z2 = a2, z > 0, if it has constant density.
> Find an equation of the tangent plane to the given parametric surface at the specified point. Graph the surface and the tangent plane. r(u, v) = (1 – u² – v²) i – vj – u k; (-1, –1, –1)
> Find an equation of the tangent plane to the given parametric surface at the specified point. Graph the surface and the tangent plane. r(u, v) = ưi + 2u sin vj + u cos vk; u = 1, v =0
> Find the flux of F (x, y, z) = sin (xyz) i + x2y j + z2ex/5 k across the part of the cylinder 4y2 + z2 = 4 that lies above the xy-plane and between the planes x = -2 and x = 2 with upward orientation. Illustrate by using a computer algebra system to draw
> Compute the outward flux of F (x, y, z) = x i + y j + z k/ (x2 + y2 + z2)3/2 through the ellipsoid 4x2 + 9y2 + 6z2 = 36.
> Find the value of ∫∫S x2y2z2 dS correct to four decimal places, where S is the part of the paraboloid z = 3 - 2x2 - y2 that lies above the xy-plane.
> Find the exact value of ∫∫S xyz dS, where S is the surface z = x2y2, 0 < x < 1, 0 < y < 2.
> Evaluate ∫∫S (x2 + y2 + z2) dS correct to four decimal places, where S is the surface z = xey, 0 < x < 1, 0 < y < 1.
> The surface with parametric equations x = 2 cos θ + r cos (θ /2) y = 2 sin θ + r cos (θ /2) z = r sins (θ /2) where -1/2 < r < 12 and 0
> Suppose S and E satisfy the conditions of the Divergence Theorem and f is a scalar function with continuous partial derivatives. Prove that These surface and triple integrals of vector functions are vectors defined by integrating each component functio
> Prove each identity, assuming that S and E satisfy the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second-order partial derivatives. (SVg – gVf) •n ds = [[[ (fV²g – gV²f) aV E
> Solve the boundary-value problem, if possible. y'' = y', y (0) = 1, y (1) = 2
> Solve the differential equation using the method of variation of parameters. y'' + 4y' + 4y = e-2x/x3
> Solve the differential equation using the method of variation of parameters. y'' – y' + y = ex/1 + x2
> Solve the differential equation using the method of variation of parameters. y'' + 3y' + 2y = sin (ex)
> Verify that the Divergence Theorem is true for the vector field F (x, y, z) = x i + y j + z k, where E is the unit ball x2 + y2 + z2 < 1.
> Solve the differential equation using the method of variation of parameters. y'' - 3y' + 2y = 1/1 + e-x
> Solve the differential equation using the method of variation of parameters. y'' + y = sec3x, 0 < x < π/2
> Solve the differential equation using the method of variation of parameters. y'' + y = sec2x, 0 < x < π/2
> Plot the vector field and guess where div F > 0 and where div F F(x, y) = (x², y²)
> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = zexy i - 3zexy j + xy k, S is the p
> Solve the differential equation using (a) undetermined coefficients and (b) variation of parameters. у" — 2у' — Зу %3х+2 3y = x + 2
> Solve the differential equation using (a) undetermined coefficients and (b) variation of parameters. 4y" + у %3 сos x
> Write a trial solution for the method of undetermined coefficients. Do not determine the coefficients. y" + 4y = ex + x sin 2.x
> Verify that the solution to Equation 1 can be written in the form x (t) = A cos (ω t + δ).
> The battery in Exercise 14 is replaced by a generator producing a voltage of E (t) = 12 sin 10t. Exercise 14: A series circuit contains a resistor with R = 24 V, an inductor with L = 2 H, a capacitor with C = 0.005 F, and a 12-V battery. (a). Find the
> Use the Divergence Theorem to calculate the surface integral ∫∫S F ∙ dS, where F (x, y, z) = x3 i + y3 j + z3 k and S is the surface of the solid bounded by the cylinder x2 + y2 = 1 and the planes z = 0 and z = 2.
> The battery in Exercise 13 is replaced by a generator producing a voltage of E (t) = 12 sin 10t. Find the charge at time t. Exercise 13: A series circuit consists of a resistor with R = 20 V, an inductor with L = 1 H, a capacitor with C = 0.002 F, and
> A series circuit contains a resistor with R = 24 V, an inductor with L = 2 H, a capacitor with C = 0.005 F, and a 12-V battery. The initial charge is Q = 0.001 C and the initial current is 0. (a). Find the charge and current at time t. (b). Graph the cha
> A series circuit consists of a resistor with R = 20 V, an inductor with L = 1 H, a capacitor with C = 0.002 F, and a 12-V battery. If the initial charge and current are both 0, find the charge and current at time t.
> The solution of the initial-value problem x2y'' + xy' + x2y = 0 y (0) = 1 y' (0) = 0 is called a Bessel function of order 0. (a). Solve the initial-value problem to find a power series expansion for the Bessel function. (b). Graph several Taylor polynomi
> Use power series to solve the differential equation. y'' + x2y' + xy = 0, y (0) = 0, y' (0) = 1
> Use power series to solve the differential equation. y'' + x2y = 0, y (0) = 1, y'(0) = 0
> Use power series to solve the differential equation. y'' – xy' - y = 0, y (0) = 1, y' (0) = 0
> Use power series to solve the differential equation. y'' = xy
> Use power series to solve the differential equation. (x – 1) y' + y' = 0
> Use power series to solve the differential equation. y'' = y
> Use Stokes’ Theorem to evaluate ∫C F ∙ dr, where F (x, y, z) = xy i + yz j + z x k, and C is the triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1), oriented counterclockwise as viewed from above.
> Use power series to solve the differential equation. y'' + xy' + y = 0
> Use power series to solve the differential equation. (x – 3) y' + 2y = 0
> Use power series to solve the differential equation. y' = x2y
> Solve the differential equation. Y'' - 6y' + 9y = 0
> Use power series to solve the differential equation. y' – y = 0
