Show that any vector field of the form F (x, y, z) = f (y,z) i + g (x,z) j + h (x,y) k where f, g and h are differentiable functions, is solenoidal
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CALCULUS; The vector field f(s)r is solenoidal for all functions of the form f(s) = C .... where C is an arbitrary constant, and only functions of this form. Please provide a detailed answer. Thanks. Let xi+ yj + zk be the position vector of a general point in 3-space and let s Irl be the length of r. Calculate the divergence of s3r. For what value of the constant k is the vector field s r solenoidal except at...
Let F 10i4u 8zk. Compute the civergence and curl of F. , div F , curl F Show steps (1 point) Let F (8y2)i(7xz)j+(6y) k Compute the following: A div F В. curl F- i+ k C, div curt F= Note: Your answers should be expressions of x, y and/or z; e.g. "3xy" or "z" or 5 (1 polnt) Consider the vector field F(r,y, ) = ( 9y , 0, -3ry) Find the divergence and curl of F div(F) VF=...
a) A vector field F is called incompressible if div F = 0. Show that a vector field of the form F = <f(y,z),g(x,z),h(x,y)> is incompressible. b) Suppose that S is a closed surface (a boundary of a solid in three dimensional space) and that F is an incompressible vector field. Show that the flux of F through S is 0. c)Show that if f and g are defined on R3 and C is a closed curve in R3 then...
please explain, not just an answer. No cursive please. Suppose that we define a function f(x) in a piecewise manner - f(z) () for x < a and f(x) = h(x) for x > a. Here, assume that g(z) and h(z) are differentiable functions. Show that f is differentiable at a if and only if f(a) g(a) and f'(a) g'(a). Suppose that we define a function f(x) in a piecewise manner - f(z) () for x a. Here, assume that...
4. Let A, X, Y, Z be normed vector spaces and B :X XY + Z be a bilinear map and f: A+X,g: A → Y be mappings that are differentiable at co E A. Show that the mapping 0 : A+Z, 2# B(f(x), g(x)) is differentiable at zo and that do (20)[h] = B(df (20)[h], g(20) + B(f(20), dg(20)[h]) (he A).
Please help out is somewhat difficult. In practice, it is often easier to show a stronger condition: if each partial derivative OJi = 1, ... , n, is continuous in a disc around p = _ (a1.... , an), then f is differentiable дх, (a1,., an) Put differently: if f is continuously differentiable at p, it is differentiable at However, just as in the one-variable case, there are functions that are differentiable but not at p = p. continuously differentiable....
4. Let A, X, Y, Z be normed vector spaces and B :X Y + Z be a bilinear map and f: A+X,9: A + Y be mappings that are differentiable at to E A. Show that the mapping 0 : A → Z, X HB(f(x), g(x)) is differentiable at Do and that dº(30)[h] = B(df (o)[N), 9(30))+ (f(x0), dg(xo)[h]) (he A).
7. (6pts) Consider F(x, y, z) = (y2 + z cos x)i + (3xy2 + 1)j + sin æk. Show that F is a conservative vector field and then compute SF. dr where C is any curve from (0,0,1) to (0,2,3).
Solve with all the steps please! Calculate the divergence and the curl of the vector field F(x,y,z) = ( x^3y)i + (xy)j + ( 213 )k. (Where Fis a vector and i,j,k stand for the standard unit vectors)
For any vector field F⃗ and any scalar function f we define a new field a) Assuming that the appropriate partial derivatives are continuous, show the following formula: b) Let ⃗x = x⃗i + y ⃗j + z ⃗k and the vector field Use the formula found in a) to answer the following question: is there a number p such that F⃗ is incompressible (that is, its divergence is zero)? f F)(x,y,z) = f(x,y,z)F(x,y, z) We were unable to transcribe...