How do you find the length of a space curve given by a vector function r (t)?
> Use Green’s first identity (Exercise 33) to prove Green’s second identity: first identity: where D and C satisfy the hypotheses of Green’s Theorem and the appropriate partial derivatives of f and
> Let r = x i + y j + z k and r = |r |. If F = r/rp, find div F. Is there a value of p for which div F = 0?
> Let r = x i + y j + z k and r = |r |. Verify each identity. (a). = r = 3 (b). = ∙ (r r) = 4r (c). ∇2r3 = 12r
> Show that if the vector field F = P i + Q j + R k is conservative and P, Q, R have continuous first-order partial derivatives, then aP aR aR ду ax az ax dz ду
> Evaluate the line integral. ∫C y dx + (x + y2) dy, C is the ellipse 4x2 + 9y2 = 36 with counterclockwise orientation
> Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F, F ∙ G, and F × G are defined by div (f F) = f div F + F â&
> Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F, F ∙ G, and F × G are defined by div (F + G) = div F + div G
> Show that any vector field of the form F (x, y, z) = f (x) i + g (y) j + h (z) k where f, t, h are differentiable functions, is irrotational.
> Is there a vector field G on R3 such that curl G =〈x, y, z〉? Explain.
> Is there a vector field G on R3 such that curl G =〈x sin y, cos y, z - xy〉? Explain.
> Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. ∫C y4 dx + 2xy3 dy, C is the ellipse x2 + 2y2 = 2
> Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. ∫C (y + e^√x) dx + (2x + cos y2) dy, C is the boundary of the region enclosed by the parabolas y = x2 and x = y2
> Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. ∫C (x2 + y2) dx + (x2 - y2) dy, C is the triangle with vertices (0, 0), (2, 1), and (0, 1)
> Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. ∫C yex dx + 2ex dy, C is the rectangle with vertices (0, 0), (3, 0), (3, 4), and (0, 4)
> Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem. ∮C x2y2 dx 1 xy dy, C consists of the arc of the parabola y = x2 from (0, 0) to (1, 1) and the line segments from (1, 1) to (0, 1) and from (0, 1) to (0, 0)
> Evaluate the line integral. ∫C yz cos x ds, C: x = t, y = 3 cos t, z = 3 sin t, 0 < t < π
> Find the work done by the force field F (x, y) = x i + (y + 2) j in moving an object along an arch of the cycloid r(t) = (t - sin t) i + (1 - cos t) j 0 < t < 2 π
> If a wire with linear density ρ (x, y) lies along a plane curve C, its moments of inertia about the x- and y-axes are defined as Find the moments of inertia for the wire in Example 3. -[v°ptx, y) ds I, = x°p(x, y) ds
> Find the mass and center of mass of a wire in the shape of the helix x = t, y = cos t, z = sin t, 0 < t < 2 π, if the density at any point is equal to the square of the distance from the origin.
> (a). Write the formulas similar to Equations 4 for the center of mass (z ̅, y ̅, z ̅) of a thin wire in the shape of a space curve C if the wire has density function ρ (x, y, z). (b). Find the center of mass of a wire in the shape of the helix x = 2 sin
> A thin wire has the shape of the first-quadrant part of the circle with center the origin and radius a. If the density function is ρ (x, y) = kxy, find the mass and center of mass of the wire.
> Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. Figure 9: F (x, y) = 0.3 i - 0.4 j ZA
> Match the functions f with the plots of their gradient vector fields labeled I–IV. Give reasons for your choices. f (x, y) = sin √ (x^2 + y^2) IV 4 -4 4 سر ه م -4
> What does Clairaut’s Theorem say?
> Complete the proof of the special case of Green’s Theorem by proving Equation 3.
> Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem. ∮C xy dx + x2y3 dy, C is the triangle with vertices (0, 0), (1, 0), and (1, 2)
> If F and G are vector fields whose component functions have continuous first partial derivatives, show that curl (F × G) = F div G - G div F + (G ∙ ∇) F – (F ∙∇) G
> Calculate ∫C F ∙ dr, where F (x, y) = 〈x2 + y, 3x - y2〉 and C is the positively oriented boundary curve of a region D that has area 6.
> Plot the gradient vector field of f together with a contour map of f. Explain how they are related to each other. f (x, y) = ln (1 + x2 + 2y2)
> Use Exercise 25 to find the moment of inertia of a circular disk of radius a with constant density ρ about a diameter. (Compare with Example 15.4.4.) Exercise 25: A plane lamina with constant density ρ (x, y) = ρ occupi
> A plane lamina with constant density ρ (x, y) = ρ occupies a region in the xy-plane bounded by a simple closed path C. Show that its moments of inertia about the axes are 1, = x* dy dx 3 3 Jc
> Use Exercise 22 to find the centroid of the triangle with vertices (0, 0), (a, 0), and (a, b), where a > 0 and b > 0. Exercise 22: Let D be a region bounded by a simple closed path C in the xy-plane. Use Green’s Theorem to prove
> Use Exercise 22 to find the centroid of a quarter-circular region of radius a. Exercise 22: Let D be a region bounded by a simple closed path C in the xy-plane. Use Green’s Theorem to prove that the coordinates of the centroid (x, y)
> Let D be a region bounded by a simple closed path C in the xy-plane. Use Green’s Theorem to prove that the coordinates of the centroid (x, y) of D are where A is the area of D. x² dy I= 2A Jc 2A
> Suppose you’re asked to determine the curve that requires the least work for a force field F to move a particle from one point to another point. You decide to check first whether F is conservative, and indeed it turns out that it is. How would you reply
> If a circle C with radius 1 rolls along the outside of the circle x2 + y2 = 16, a fixed point P on C traces out a curve called an epicycloid, with parametric equations x = 5 cos t - cos 5t, y = 5 sin t - sin 5t. Graph the epicycloid and use (5) to find t
> Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem. ∮C y dx - x dy, C is the circle with center the origin and radius 4
> Show that there is no vector field G such that curl G = 2x i + 3yz j - xz2 k
> Show that the line integral is independent of path and evaluate the integral. ∫C 2xe-y dx + (2y - x2e-y) dy, C is any path from (1, 0) to (2, 1)
> (a) Find a function f such that F = ∇f and (b) use part (a) to evaluate ∫C F ∙ dr along the given curve C. 12. F (x, y, z) = yz i + xz j + (xy + 2z) k, C is the line segment from (1, 0, -2) to (4, 6, 3)
> Use Green’s Theorem to evaluate ∫C F ∙ dr. (Check the orientation of the curve before applying the theorem.) F (x, y) =〈√ (x^2 + 1), tan^ (-1) x〉, C is the triangle from (0, 0) to (1, 1) to (0, 1) to (0, 0)
> Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem. ∮C y2 dx + x2y dy, C is the rectangle with vertices (0, 0), (5, 0), (5, 4), and (0, 4)
> Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f. F (x, y) = (y2 cos x + cos y) i + (2y sin x - x sin y) j
> Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f. F (x, y) = (2xy + y-2) i + (x2 - 2xy-3) j, y > 0
> Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f. F (x, y) = (yex + sin y) i + (ex + x cos y) j
> Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f. F (x, y) = yex i + (ex + ey) j
> Let f be a joint density function of a pair of continuous random variables X and Y. (a). Write a double integral for the probability that X lies between a and b and Y lies between c and d. (b). What properties does f possess? (c). What are the expected v
> If a lamina occupies a plane region D and has density function ρ (x, y), write expressions for each of the following in terms of double integrals. (a). The mass (b). The moments about the axes (c). The center of mass (d). The moments of inertia about the
> Find curl F and div F if F (x, y, z) = e-x sin y i + e-y sin z j + e-z sin x k
> Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f. F (x, y) = (xy + y2) i + (x2 + 2xy) j
> (a). What is the definition of curvature? (b). Write a formula for curvature in terms of r'(t) and T'(t). (c). Write a formula for curvature in terms of r'(t) and r''(t). (d). Write a formula for the curvature of a plane curve with equation y - f (x).
> Find the gradient vector field ∆f of f and sketch it. f (x, y) = 1/2 (x2 - y2)
> Find the gradient vector field ∆f of f and sketch it. f (x, y) = ½ (x – y)2
> Find the work done by the force field F in moving an object from P to Q. F (x, y) = (2x + y) i + x j; P (1, 1), Q (4, 3)
> Find the work done by the force field F in moving an object from P to Q. F (x, y) = x3 i + y3 j; P (1, 0), Q (2, 2)
> Suppose an experiment determines that the amount of work required for a force field F to move a particle from the point (1, 2) to the point (5, -3) along a curve C1 is 1.2 J and the work done by F in moving the particle along another curve C2 between the
> If u and v are differentiable vector functions, c is a scalar, and f is a real-valued function, write the rules for differentiating the following vector functions. (a). u (t) + v (t) (b). cu (t) (c). f (t) u (t) (d). u(t) ∙ v (t) (e). u (t) × v (t) (
> A table of values of a function f with continuous gradient is given. Find ∫C ∇f dr, where C has parametric equations x = t2 + 1 y = t3 + t 0 < t < 1
> Use Green’s Theorem to evaluate ∫C x2y dx - xy2 dy, where C is the circle x2 + y2 = 4 with counterclockwise orientation.
> (a) Find a function f such that F = ∇f and (b) use part (a) to evaluate ∫C F ∙ dr along the given curve C. 12. F(x, y, z) = sin y i+ (x cos y + cos z)j – y sin z k, C: r(t) = sin ti + tj+ 2t k,
> (a) Find a function f such that F = ∇f and (b) use part (a) to evaluate ∫C F ∙ dr along the given curve C. 12. F(x, y, z) = yze" i + e* j+ xye* k, C: r(t) = (t² + 1) i + (t² – 1) j + (r² - 21)
> (a) Find a function f such that F = ∇f and (b) use part (a) to evaluate ∫C F ∙ dr along the given curve C. 12. F(x, y, z) = (y²z + 2xz²) i + 2xyz j + (xy² + 2x²z) k, C: x= Vī, y = t + 1, z = t²
> (a) Find a function f such that F = ∇f and (b) use part (a) to evaluate ∫C F ∙ dr along the given curve C. 12. F (x, y) = (1 + xy) exy i + x2exy j, C: r (t) = cos t i + 2 sin t j, 0 < t < π/2
> (a) Find a function f such that F = ∆f and (b) use part (a) to evaluate ∫C F ∙ dr along the given curve C. 12. F (x, y) = x2y3 i + x3y2 j, C: r (t) =〈t3 - 2t, t3 + 2t〉, 0 < t < 1
> (a) Find a function f such that F = ∇f and (b) use part (a) to evaluate ∫C F ∙ dr along the given curve C. 12. F (x, y) = (3 + 2xy2) i + 2x2y j, C is the arc of the hyperbola y = 1/x from (1, 1) to (4, 1/4)
> Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. Figure 9: F (x, y) = -1/2 i + (y – x) j ZA
> Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f. F (x, y) = (ln y + y/x) i + (ln x + x/y) j
> What is a vector function? How do you find its derivative and its integral?
> Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. Figure 9: F (x, y) = 1/2 x i + y j ZA
> Use Green’s Theorem to evaluate ∫C √1 + x3 dx + 2xy dy where C is the triangle with vertices (0, 0), (1, 0), and (1, 3).
> Suppose a solid object occupies the region E and has density function ρ (x, y, z). Write expressions for each of the following. (a). The mass (b). The moments about the coordinate planes (c). The coordinates of the center of mass (d). The moments of iner
> Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. Figure 9: F (x, y, z) = i ZA
> Write an expression for the area of a surface with equation z = f (x, y), (x, y) ∈ D.
> How do you find the tangent vector to a smooth curve at a point? How do you find the tangent line? The unit tangent vector?
> (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.
