UD CALC (241 ONLY) W/1 TERM ACCESS >IB
UD CALC (241 ONLY) W/1 TERM ACCESS >IB
8th Edition
ISBN: 9781337051545
Author: Stewart
Publisher: CENGAGE C
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Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …
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Chapter 16.P, Problem 5P

Prove the following identity:

∇ ( F ⋅ G ) = ( F ⋅ ∇ ) G+ ( G ⋅ ∇ ) F + F × curlG+G × curlF

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Rama/Shutterstock.com Romaset/Shutterstock.com The power station has three different hydroelectric turbines, each with a known (and unique) power function that gives the amount of electric power generated as a function of the water flow arriving at the turbine. The incoming water can be apportioned in different volumes to each turbine, so the goal of this project is to determine how to distribute water among the turbines to give the maximum total energy production for any rate of flow. Using experimental evidence and Bernoulli's equation, the following quadratic models were determined for the power output of each turbine, along with the allowable flows of operation: 6 KW₁ = (-18.89 +0.1277Q1-4.08.10 Q) (170 - 1.6 · 10¯*Q) KW2 = (-24.51 +0.1358Q2-4.69-10 Q¹²) (170 — 1.6 · 10¯*Q) KW3 = (-27.02 +0.1380Q3 -3.84-10-5Q) (170 - 1.6-10-ºQ) where 250 Q1 <1110, 250 Q2 <1110, 250 <3 < 1225 Qi = flow through turbine i in cubic feet per second KW = power generated by turbine i in kilowatts
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Chapter 16 Solutions

UD CALC (241 ONLY) W/1 TERM ACCESS >IB

Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - Prob. 7ECh. 16.1 - Prob. 8ECh. 16.1 - Prob. 9ECh. 16.1 - Prob. 10E
Ch. 16.1 - Prob. 11ECh. 16.1 - Match the vector fields F with the plots labelled...Ch. 16.1 - Match the vector fields F with the plots labelled...Ch. 16.1 - Match the vector fields F with the plots labeled...Ch. 16.1 - Match the vector fields F on 3 with the plots...Ch. 16.1 - Match the vector fields F on 3 with the plots...Ch. 16.1 - Match the vector fields F on 3 with the plots...Ch. 16.1 - Match the vector fields F on 3 with the plots...Ch. 16.1 - Prob. 19ECh. 16.1 - Prob. 20ECh. 16.1 - Prob. 21ECh. 16.1 - Find the gradient vector field of f. f(s,t)=2s+3tCh. 16.1 - Find the gradient vector field of f....Ch. 16.1 - Find the gradient vector field of f....Ch. 16.1 - Find the gradient vector field f of f and sketch...Ch. 16.1 - Find the gradient vector field f of f and sketch...Ch. 16.1 - Prob. 27ECh. 16.1 - Plot the gradient vector field of f together with...Ch. 16.1 - Match the functions f with the plots of their...Ch. 16.1 - Match the functions f with the plots of their...Ch. 16.1 - Match the functions f with the plots of their...Ch. 16.1 - Match the functions f with the plots of their...Ch. 16.1 - A particle moves in a velocity field...Ch. 16.1 - Prob. 34ECh. 16.1 - The flow lines or streamlines of a vector field...Ch. 16.1 - a Sketch the vector field F(x,y)=i+xj and then...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Prob. 4ECh. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Prob. 14ECh. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Prob. 16ECh. 16.2 - Let F be the vector fields shown in the figure. a...Ch. 16.2 - The figure shows a vector field F and two curves...Ch. 16.2 - Prob. 19ECh. 16.2 - Prob. 20ECh. 16.2 - Prob. 21ECh. 16.2 - Evaluate the line integral CFdr, where C is given...Ch. 16.2 - Prob. 23ECh. 16.2 - Prob. 24ECh. 16.2 - Prob. 25ECh. 16.2 - Prob. 26ECh. 16.2 - Use a graph of the vector field F and the curve C...Ch. 16.2 - Use a graph of the vector field F and the curve C...Ch. 16.2 - a Evaluate the line integral CFdr, where...Ch. 16.2 - a Evaluate the line integral CFdr, where...Ch. 16.2 - Find the exact value of Cx3y3zds, where C is the...Ch. 16.2 - a Find the work done by the force field...Ch. 16.2 - A thin wire is bent into the shape of a semicircle...Ch. 16.2 - A thin wire has the shape of the first-quadrant...Ch. 16.2 - a Write the formulas similar to Equations 4 for...Ch. 16.2 - Find the mass and center of mass of a wire in the...Ch. 16.2 - Prob. 37ECh. 16.2 - Prob. 38ECh. 16.2 - Find the work done by the force field...Ch. 16.2 - Find the work done by the force field...Ch. 16.2 - Prob. 41ECh. 16.2 - Prob. 42ECh. 16.2 - Prob. 43ECh. 16.2 - An object with mass m moves with position function...Ch. 16.2 - A 160-lb man carries a 25-lb can of paint up a...Ch. 16.2 - Prob. 46ECh. 16.2 - a Show that a constant force field does zero work...Ch. 16.2 - Prob. 48ECh. 16.2 - If C is a smooth curve given by a vector function...Ch. 16.2 - Prob. 50ECh. 16.2 - An object moves along the curve C shown in the...Ch. 16.2 - Experiments show that a steady current I in a long...Ch. 16.3 - The figure shows a curve C and a contour map of a...Ch. 16.3 - Prob. 2ECh. 16.3 - Determine whether or not F is a conservative...Ch. 16.3 - Determine whether or not F is a conservative...Ch. 16.3 - Determine whether or not F is a conservative...Ch. 16.3 - Prob. 6ECh. 16.3 - Prob. 7ECh. 16.3 - Prob. 8ECh. 16.3 - Prob. 9ECh. 16.3 - Determine whether or not F is a conservative...Ch. 16.3 - The figure shows the vector field F(x,y)=2xy,x2...Ch. 16.3 - a Find a function f such that F=f and b use part a...Ch. 16.3 - a Find a function f such that F=f and b use part a...Ch. 16.3 - a Find a function f such that F=f and b use part a...Ch. 16.3 - a Find a function f such that F=f and b use part a...Ch. 16.3 - Prob. 16ECh. 16.3 - Prob. 17ECh. 16.3 - Prob. 18ECh. 16.3 - Show that the line integral is independent of path...Ch. 16.3 - Show that the line integral is independent of path...Ch. 16.3 - Suppose youre asked to determine the curve that...Ch. 16.3 - Prob. 22ECh. 16.3 - Find the work done by the force field F in moving...Ch. 16.3 - Find the work done by the force field F in moving...Ch. 16.3 - Is the vector field shown in the figure...Ch. 16.3 - Is the vector field shown in the figure...Ch. 16.3 - If F(x,y)=sinyi+(1+xcosy)j, use a plot to guess...Ch. 16.3 - Let F=f, where f(x,y)=sin(x2y). Find curves C1 and...Ch. 16.3 - Show that if the vector field F=Pi+Qj+Rk is...Ch. 16.3 - Use Exercise 29 to show that the line integral...Ch. 16.3 - Determine whether or not the given set is a open,...Ch. 16.3 - Determine whether or not the given set is a open,...Ch. 16.3 - Determine whether or not the given set is a open,...Ch. 16.3 - Prob. 34ECh. 16.3 - Prob. 35ECh. 16.3 - a Suppose that F is an inverse square force field,...Ch. 16.4 - Evaluate the line integral by two methods: a...Ch. 16.4 - Evaluate the line integral by two methods: a...Ch. 16.4 - Evaluate the line integral by two methods: a...Ch. 16.4 - Evaluate the line integral by two methods: a...Ch. 16.4 - Use Greens Theorem to evaluate the line integral...Ch. 16.4 - Use Greens Theorem to evaluate the line integral...Ch. 16.4 - Use Greens Theorem to evaluate the line integral...Ch. 16.4 - Use Greens Theorem to evaluate the line integral...Ch. 16.4 - Use Greens Theorem to evaluate the line integral...Ch. 16.4 - Use Greens Theorem to evaluate the line integral...Ch. 16.4 - Use Greens Theorem to evaluate cFdr. Check the...Ch. 16.4 - Prob. 12ECh. 16.4 - Prob. 13ECh. 16.4 - Prob. 14ECh. 16.4 - Verify Greens Theorem by using a computer algebra...Ch. 16.4 - Verify Greens Theorem by using a computer algebra...Ch. 16.4 - Prob. 17ECh. 16.4 - A particle starts at the origin, moves along the...Ch. 16.4 - Prob. 19ECh. 16.4 - If a circle C with radius 1 rolls along the...Ch. 16.4 - Prob. 21ECh. 16.4 - Prob. 22ECh. 16.4 - Use Exercise 22 to find the centroid of a...Ch. 16.4 - Prob. 24ECh. 16.4 - A plane lamina with constant density (x,y)=...Ch. 16.4 - Prob. 26ECh. 16.4 - Use the method of Example 5 to calculate CFdr,...Ch. 16.4 - Prob. 28ECh. 16.4 - If F is the vector field of Example 5, show that...Ch. 16.4 - Complete the proof of the special case of Greens...Ch. 16.4 - Prob. 31ECh. 16.5 - Find a the curl and b the divergence of the vector...Ch. 16.5 - Find a the curl and b the divergence of the vector...Ch. 16.5 - Find a the curl and b the divergence of the vector...Ch. 16.5 - Find a the curl and b the divergence of the vector...Ch. 16.5 - Find a the curl and b the divergence of the vector...Ch. 16.5 - Find a the curl and b the divergence of the vector...Ch. 16.5 - Find a the curl and b the divergence of the vector...Ch. 16.5 - Find a the curl and b the divergence of the vector...Ch. 16.5 - The vector field F is shown in the xy-plane and...Ch. 16.5 - The vector field F is shown in the xy-plane and...Ch. 16.5 - The vector field F is shown in the xy-plane and...Ch. 16.5 - Let f be a scalar field and F a vector field....Ch. 16.5 - Determine whether or not the vector field is...Ch. 16.5 - Determine whether or not the vector field is...Ch. 16.5 - Prob. 15ECh. 16.5 - Determine whether or not the vector field is...Ch. 16.5 - Prob. 17ECh. 16.5 - Determine whether or not the vector field is...Ch. 16.5 - Is there a vector field G on 3 such that curl...Ch. 16.5 - Prob. 20ECh. 16.5 - Show that any vector field of the form...Ch. 16.5 - Prob. 22ECh. 16.5 - Prob. 23ECh. 16.5 - Prove the identity, assuming that the appropriate...Ch. 16.5 - Prob. 25ECh. 16.5 - Prove the identity, assuming that the appropriate...Ch. 16.5 - Prob. 27ECh. 16.5 - Prove the identity, assuming that the appropriate...Ch. 16.5 - Prove the identity, assuming that the appropriate...Ch. 16.5 - Let r=xi+yj+zk and r=|r|. Verify each identity. a...Ch. 16.5 - Let r=xi+yj+zk and r=|r|. Verify each identity. a...Ch. 16.5 - Let r=xi+yj+zk and r=|r|. If F=r/rp, find div F....Ch. 16.5 - Use Greens Theorem in the form of Equation 13 to...Ch. 16.5 - Prob. 34ECh. 16.5 - Recall from Section 14.3 that a function g is...Ch. 16.5 - Prob. 36ECh. 16.5 - This exercise demonstrates a connection between...Ch. 16.5 - Maxwells equations relating the electric field E...Ch. 16.5 - We have seen that all vector fields of the form...Ch. 16.6 - Determine whether the points P and Q lie on the...Ch. 16.6 - Determine whether the points P and Q lie on the...Ch. 16.6 - Prob. 3ECh. 16.6 - Identify the surface with the given vector...Ch. 16.6 - Prob. 5ECh. 16.6 - Identify the surface with the given vector...Ch. 16.6 - Use a computer to graph the parametric surface....Ch. 16.6 - Prob. 8ECh. 16.6 - Prob. 9ECh. 16.6 - Prob. 10ECh. 16.6 - Prob. 11ECh. 16.6 - Prob. 12ECh. 16.6 - Match the equations with the graphs labeled IVI...Ch. 16.6 - Match the equations with the graphs labeled IVI...Ch. 16.6 - Match the equations with the graphs labeled IVI...Ch. 16.6 - Match the equations with the graphs labeled IVI...Ch. 16.6 - Prob. 17ECh. 16.6 - Prob. 18ECh. 16.6 - Find da parametric representation for the surface....Ch. 16.6 - Find da parametric representation for the surface....Ch. 16.6 - Find da parametric representation for the surface....Ch. 16.6 - Find da parametric representation for the surface....Ch. 16.6 - Find da parametric representation for the surface....Ch. 16.6 - Find da parametric representation for the surface....Ch. 16.6 - Find da parametric representation for the surface....Ch. 16.6 - Find da parametric representation for the surface....Ch. 16.6 - Prob. 27ECh. 16.6 - Prob. 28ECh. 16.6 - Prob. 29ECh. 16.6 - Prob. 30ECh. 16.6 - a What happens to the spiral tube in Example 2 see...Ch. 16.6 - Prob. 32ECh. 16.6 - Find an equation of the tangent plane to the given...Ch. 16.6 - Find an equation of the tangent plane to the given...Ch. 16.6 - Prob. 35ECh. 16.6 - Prob. 36ECh. 16.6 - Prob. 37ECh. 16.6 - Prob. 38ECh. 16.6 - Prob. 39ECh. 16.6 - Prob. 40ECh. 16.6 - Find the area of the surface. The part of the...Ch. 16.6 - Find the area of the surface. The part of the cone...Ch. 16.6 - Find the area of the surface. The surface...Ch. 16.6 - Find the area of the surface. The part of the...Ch. 16.6 - Find the area of the surface. The part of the...Ch. 16.6 - Find the area of the surface. The part of the...Ch. 16.6 - Find the area of the surface. The part of the...Ch. 16.6 - Prob. 48ECh. 16.6 - Find the area of the surface. The surface with...Ch. 16.6 - Find the area of the surface. The part of the...Ch. 16.6 - Prob. 51ECh. 16.6 - Prob. 52ECh. 16.6 - Prob. 53ECh. 16.6 - Prob. 54ECh. 16.6 - Prob. 55ECh. 16.6 - Prob. 56ECh. 16.6 - Prob. 57ECh. 16.6 - Prob. 58ECh. 16.6 - a Show that the parametric equations...Ch. 16.6 - a Show that the parametric equations...Ch. 16.6 - Find the area of the part of the sphere...Ch. 16.6 - The figure shows the surface created when the...Ch. 16.6 - Prob. 63ECh. 16.6 - a Find a parametric representation for the torus...Ch. 16.7 - Let S be the surface of the box enclosed by the...Ch. 16.7 - Prob. 2ECh. 16.7 - Prob. 3ECh. 16.7 - Suppose that f(x,y,z)=g(x2+y2+z2), where g is a...Ch. 16.7 - Evaluate the surface integral. S(x+y+z)dS, S is...Ch. 16.7 - Evaluate the surface integral. SxyzdS, S is the...Ch. 16.7 - Prob. 7ECh. 16.7 - Evaluate the surface integral. S(x2+y2)dS, S is...Ch. 16.7 - Evaluate the surface integral. Sx2yzdS, S is the...Ch. 16.7 - Prob. 10ECh. 16.7 - Evaluate the surface integral. SxdS, S is the...Ch. 16.7 - Evaluate the surface integral. SydS, S is the...Ch. 16.7 - Evaluate the surface integral. Sz2dS, S is the...Ch. 16.7 - Evaluate the surface integral. Sy2z2dS, S is the...Ch. 16.7 - Prob. 15ECh. 16.7 - Evaluate the surface integral. Sy2dS, S is the...Ch. 16.7 - Prob. 17ECh. 16.7 - Evaluate the surface integral. S(x+y+z)dS, S is...Ch. 16.7 - Evaluate the surface integral. SxzdS, S is the...Ch. 16.7 - Prob. 20ECh. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Prob. 33ECh. 16.7 - Prob. 34ECh. 16.7 - Prob. 35ECh. 16.7 - Find the flux of F(x,y,z)=sin(xyz)i+x2yj+z2ex/5k...Ch. 16.7 - Prob. 37ECh. 16.7 - Prob. 38ECh. 16.7 - Find the centre of mass of the hemisphere...Ch. 16.7 - Find the mass of a thin funnel in the shape of a...Ch. 16.7 - Prob. 41ECh. 16.7 - Let S be the part of the sphere x2+y2+z2=25 that...Ch. 16.7 - Prob. 43ECh. 16.7 - Prob. 44ECh. 16.7 - Use Gausss Law to find the charge contained in the...Ch. 16.7 - Prob. 46ECh. 16.7 - Prob. 47ECh. 16.7 - Prob. 48ECh. 16.7 - Prob. 49ECh. 16.8 - A hemisphere H and a portion P of a paraboloid are...Ch. 16.8 - Use Stokes Theorem to evaluate ScurlFdS...Ch. 16.8 - Use Stokes Theorem to evaluate ScurlFdS....Ch. 16.8 - Use Stokes Theorem to evaluate ScurlFdS....Ch. 16.8 - Use Stokes Theorem to evaluate ScurlFdS....Ch. 16.8 - Use Stokes Theorem to evaluate ScurlFdS...Ch. 16.8 - Use Stokes Theorem to evaluate cFdr. In each case...Ch. 16.8 - Prob. 8ECh. 16.8 - Use Stokes Theorem to evaluate cFdr. In each case...Ch. 16.8 - Use Stokes Theorem to evaluate cFdr. In each case...Ch. 16.8 - a Use Stokes Theorem to evaluate cFdr, where...Ch. 16.8 - a Use Stokes Theorem to evaluate cFdr, where...Ch. 16.8 - Verify the Stokes Theorem is true for the given...Ch. 16.8 - Verify that Stokes Theorem is true for given...Ch. 16.8 - Verify that Stokes Theorem is true for given...Ch. 16.8 - Let C be a simple closed smooth curve that lies in...Ch. 16.8 - A particle moves along line segments from the...Ch. 16.8 - Evaluate C(y+sinx)dx+(z2+cosy)dy+x3dz where C is...Ch. 16.8 - Prob. 19ECh. 16.8 - Suppose S and C satisfy the hypotheses of Stokes...Ch. 16.9 - Verify that the Divergence Theorem is true for the...Ch. 16.9 - Verify that the Divergence Theorem is true for the...Ch. 16.9 - Prob. 3ECh. 16.9 - Prob. 4ECh. 16.9 - Prob. 5ECh. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Prob. 7ECh. 16.9 - Prob. 8ECh. 16.9 - Prob. 9ECh. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Prob. 14ECh. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Use a computer algebra system to plot the vector...Ch. 16.9 - Use a Divergence Theorem to evaluate SFdS, where...Ch. 16.9 - Let F(x,y,z)=ztan1(y2)i+z3ln(x2+1)j+zk. Find the...Ch. 16.9 - A vector field F is shown. Use the interpretation...Ch. 16.9 - a Are the points P1 and P2 sources or sinks for...Ch. 16.9 - Prob. 21ECh. 16.9 - Prob. 22ECh. 16.9 - Verify that div E=0 for the electric field...Ch. 16.9 - Prob. 24ECh. 16.9 - Prob. 25ECh. 16.9 - Prove each identity, assuming that S and E satisfy...Ch. 16.9 - Prove each identity, assuming that S and E satisfy...Ch. 16.9 - Prove each identity, assuming that S and E satisfy...Ch. 16.9 - Prove each identity, assuming that S and E satisfy...Ch. 16.9 - Prob. 30ECh. 16.9 - Suppose S and E satisfy the conditions of the...Ch. 16.9 - Prob. 32ECh. 16.R - Prob. 1CCCh. 16.R - a What is a conservative vector field? b What is...Ch. 16.R - Prob. 3CCCh. 16.R - a Define the line integral of a vector field F...Ch. 16.R - Prob. 5CCCh. 16.R - Prob. 6CCCh. 16.R - Prob. 7CCCh. 16.R - Write expressions for the area enclosed by a curve...Ch. 16.R - Prob. 9CCCh. 16.R - Prob. 10CCCh. 16.R - Prob. 11CCCh. 16.R - Prob. 12CCCh. 16.R - Prob. 13CCCh. 16.R - Prob. 14CCCh. 16.R - Prob. 15CCCh. 16.R - Prob. 16CCCh. 16.R - Prob. 1TFQCh. 16.R - Prob. 2TFQCh. 16.R - Prob. 3TFQCh. 16.R - Prob. 4TFQCh. 16.R - Prob. 5TFQCh. 16.R - Prob. 6TFQCh. 16.R - Prob. 7TFQCh. 16.R - Prob. 8TFQCh. 16.R - Prob. 9TFQCh. 16.R - Prob. 10TFQCh. 16.R - Prob. 11TFQCh. 16.R - Prob. 12TFQCh. 16.R - Prob. 13TFQCh. 16.R - A vector field F, a curve C, and a point P are...Ch. 16.R - Evaluate the line integral. cxds, C is the arc of...Ch. 16.R - Evaluate the line integral. cyzcosxds,...Ch. 16.R - Evaluate the line integral. cydx+(x+y2)dy, C is...Ch. 16.R - Prob. 5ECh. 16.R - Evaluate the line integral. cxydx+eydy+xzdz, C is...Ch. 16.R - Prob. 7ECh. 16.R - Evaluate the line integral. cFdr, where...Ch. 16.R - Prob. 9ECh. 16.R - Find the work done by the force field...Ch. 16.R - Show that F is a conservative vector field. Then...Ch. 16.R - Prob. 12ECh. 16.R - Prob. 13ECh. 16.R - Show that F is a conservative and use this fact to...Ch. 16.R - Verify that Greens Theorem is true for the line...Ch. 16.R - Prob. 16ECh. 16.R - Use Greens theorem to evaluate cx2ydxxy2dy, where...Ch. 16.R - Prob. 18ECh. 16.R - Show that there is no vector field G such that...Ch. 16.R - Prob. 20ECh. 16.R - Prob. 21ECh. 16.R - If f and g are twice differentiable functions,...Ch. 16.R - If f is a harmonic function, that is, 2f=0, show...Ch. 16.R - a Sketch the curve C with parametric equations...Ch. 16.R - Prob. 25ECh. 16.R - Prob. 26ECh. 16.R - Prob. 27ECh. 16.R - Prob. 28ECh. 16.R - Evaluate the surface integral. sFdS, where...Ch. 16.R - Prob. 30ECh. 16.R - Verify that Stokes Theorem is true for the vector...Ch. 16.R - Prob. 32ECh. 16.R - Use Stokes Theorem to evaluate cFdr, where...Ch. 16.R - Use the Divergence Theorem to calculate the...Ch. 16.R - Prob. 35ECh. 16.R - Compute the outward flux of...Ch. 16.R - Prob. 37ECh. 16.R - Let F(x,y)=(2x3+2xy22y)i+(2y3+2x2y+2x)jx2+y2...Ch. 16.R - Find sFndS, where F(x,y,z)=xi+yj+zk and S is the...Ch. 16.R - Prob. 40ECh. 16.R - Prob. 41ECh. 16.P - Let S be a smooth parametric surface and P be a...Ch. 16.P - Find the positively oriented simple closed curve C...Ch. 16.P - Let C be a simple closed piecewise-smooth space...Ch. 16.P - Investigate the shape of the surface with...Ch. 16.P - Prove the following identity:...Ch. 16.P - The depicts the sequence of events in each...
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