Evaluating a Line Integral of a Vector Field In Exercises 29-34, evaluate ∫ c F ⋅ d r . F ( x , y , z ) = x y i + x z j + y z k C : r ( t ) = t i + t 2 j + 2 t k , 0 ≤ t ≤ 1
Evaluating a Line Integral of a Vector Field In Exercises 29-34, evaluate ∫ c F ⋅ d r . F ( x , y , z ) = x y i + x z j + y z k C : r ( t ) = t i + t 2 j + 2 t k , 0 ≤ t ≤ 1
Evaluating a Line Integral of a Vector Field In Exercises 29-34, evaluate
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Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Flux of a vector field?
Let S be a closed surface consisting of a paraboloid (z = x²+y²), with (0≤z≤1), and capped by the disc (x²+y² ≤1) on the plane (z=1). Determine the flow of the vector field F (x,y,z) = zj − yk, in the direction that points out across the surface S.
a. Show that the outward flux of the position vector field F = x i + y j + z k through a smooth closed surface S is three times the volume of the region enclosed by the surface.
b. Let n be the outward unit normal vector field on S. Show that it is not possible for F to be orthogonal to n at every point of S
Using Green's Theorem on this vector field problem, compute a) the circulation on the boundary of R in terms of a and b, and b) the outward flux across the boundary of R in terms of a and b.
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