Evaluating a Line Integral of a
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Calculus
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- Line integrals of vector fields in the plane Given the followingvector fields and oriented curves C, evaluate ∫C F ⋅ T ds. F = ⟨-y, x⟩ on the parabola y = x2 from (0, 0) to (1, 1)arrow_forwardThe figure shows a vector field F and two curves C_1 and C_2. Are the line integrals of F over C_1 and C_2 positive, negative, or zero? Explain.arrow_forwardcheck the stokes theorem for vactor field A=(x)i+(y)j+(2xy)k where S is the lower hemisphere x2+y2+z2=4 and z<=0arrow_forward
- Consider the vector field ?(?,?,?)=(?+?)?+(2?+?)?+(2?+?)? F ( x , y , z ) = ( z + y ) i + ( 2 z + x ) j + ( 2 y + x ) k . a) Find a function ? f such that ?=∇? F = ∇ f and ?(0,0,0)=0 f ( 0 , 0 , 0 ) = 0 . ?(?,?,?)= f ( x , y , z ) = b) Suppose C is any curve from (0,0,0) ( 0 , 0 , 0 ) to (1,1,1). ( 1 , 1 , 1 ) . Use part a) to compute the line integral ∫??⋅?? ∫ C F ⋅ d r .arrow_forwardUsing Gauss' theorem to calculate the flow of the vector field 3x3 F: F (x, y, z) = (x^2z, 2x^2, 3z^2) exiting the cylinder defined from the relations x ^2+y ^2<=1, 1<= z <= 2.arrow_forwardDetermine whether the vector field is conservative. If it is, find a potential function for the vector field. (If the vector field is not conservative, enter DNE.) F(x,y)= -y/x2 i +1/xj f(x,y)= __?_ + Carrow_forward
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