Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x² + y² = 81, 0 ≤ z < 1, and a hemisphericalcap defined by x² + y² + (z − 1)² = 81, z > 1. For the vector field F = (zx + z²y+7y, z³yx + 4x, z²x²), computeany way you like.SM(VF) dS=(▼ × F) dS in

Given that M be the capped cylindrical surface which is the unionof two surfaces, a cylinder given…

Answered: Let M be the capped cylindrical surface…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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A vector field representing wind speeds is given by v = ry³ i + 3x³y³j. A walker can take one of two paths between points A(0,0) and B(1,1). Determine which path has more work done by the field? a) The first path follows y = x4. Find the work done by the field following this path. For path c1, v • dr =( )dr Sa v• dr = Note: Show solution to integral as answer rounded to 2 decimal places. b) The second path follows y = x. Find the work done by the field following this path. For path c2, v • dr =( )dx S,v• dr Note: Show solution to integral as answer rounded to 2 decimal places. c) Which path should you take if you want to put in less effort? O Path on curve 1 O Path on curve 2 O Either path, effort is the same.
sketch properly and provide full solutions! will give thumbs up!
Consider inner-product =| f(x)g(x) dx defined for vector space C[-1, 1] , then for the function f(x) = 3 x the inner-product equals: -1 Select one: а. 6 O b. 3 О с. 1 O d. zero
A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at each point by w(x, y, z) = 25 − 4(x² + y² + z²) °C. Use the fact that heat flow is given by the vector field F = -KVw and the rate of heat flow across a surface S within the solid is given by -K Vw ds. Find the rate of heat flow out of a sphere of radius 1 (centered at the origin) inside a large cube of copper (K = 400 kW/(m - K)). (Use symbolic notation and fractions where needed.) x J[, VW S -K Incorrect VwdS= 12800 kW
A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at each point by w(x, y, z) = 15 - 4(x² + y² + z²) °C. Use the fact that heat flow is given by the vector field F = -KVw and the rate of heat flow across a surface S within the solid is given by -K , Vw ds. Find the rate of heat flow out of a sphere of radius 1 (centered at the origin) inside a large cube of copper (K = 400 kW/(mK)). (Use symbolic notation and fractions where needed.) -K Incorrect 1₁² VwdS= 12800T 32 Sille Jour me que an kW
4. Assume A is a constant vector field and R = xi + yi + zk. (a) Simplify the expression (R. VR) · Ã+ V ·V × V × ((Ã × R) × (R × Ã)). (b) Assuming A the position vector of a point on the unit sphere centered at the origin, evaluate the above expression at R = A.
Q5: a) Show that the vector field F = (x² - y² + x)i- (2xy + y)j, is irrational.
G is a vector field defined as G = Bxlnzi + Dy zj + (+ y³) k. b) If C is the straight line from (1,1,1) to (2,1,2) find S, 2xlnzdx + 2y²zdy +y dz.
In Cartesian coordinates, a vector field takes the form F= 2xzi+ 2yzj+ (X2+y²)k a. State whether F is conservative, and give a one-sentence justification for your statement b. Calculate the line integral of F along a straight-line path starting at the origin and ending at the point (a, b, c). This path has the parametric representation x=at, y=bt, z=ct (0sts1). c. Given that the point (a, b, c) could be anywhere, use your answer to part (b) to find the scalar potential function U(x, y, z) corresponding to F, such that F=-VU.

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