If ⃗v = √x⃗i+y⃗j+z⃗k x2+y 2+z 2 , find the value of div ⃗v.
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Find the directional derivative of x 2 y 2 z 2 at the point (1, 1, −1) in the direction of the tangent to the curve x = e t , y = sin 2t + 1, z = 1 − cost at t = 0.
Find the constants m and n such that the surface mx2 − 2nyz = (m + 4)x will be orthogonal to the surface 4x 2 y + z 3 = 4 at the point (1, −1, 2).
Find an equation of the line that is tangent to the graph of f and parallel to the given line.function: f(x) = x2 − 6line: 2x + y = 0
number 4 parts a b and c The symbols Vf, V.. bols vf, V az oy F, and V x F are defined by Of = grad f 7. F = div F VXF = curl F After eq. (3.12) is memorized, formulas (3.13) through (3.15) venient ways of remembering the expressions for gradient, diverge just operate with V as though it were a vector. Henceforth, we breviations frequently. EXERCISES 1. If f(x,y,z) = x²y + z, what is f(2,3,4)?...
3. Consider the functions \(f(x, y, z)=x y z\) and \(\mathbf{F}(x, y, z)=y z^{2} i+x^{2} z j+x y^{2} k\). Determine which of the following operations can be carried out and find its value:div \(f, \operatorname{grad} f,\) div \(\mathbf{F},\) curl div \(\mathbf{F}\) and div curl \(\mathbf{F}\).
(x2 + y2 + z?)1/2, and e, = r1(x, y, z) is the unit radial vector. Let F = r"e, where n is any number, r= (a) Calculate div(F). (2+n)"-1 (b) Calculate the flux of F through the surface of a sphere of radius R centered at the origin. 4TR"+2 F. ds, where C is a closed curve that does not pass through the origin? (c) What is the value of (d) A function o satisfying Ap = 0 is...
ems (1 point) A) Consider the vector field F(x, y, z) = (6yz, -7zz, zy). Find the divergence and curl of F. div(F) = V.F= curl(F) = V F =( ). 5 (5x?, 2(x + y), -7(x + y + x)) 7 B) Consider the vector field F(x, y, z) Find the divergence and curl of F. div(F) = V.P= curl(F) = V XF =( 8 9 10 )
2. Evaluate the surface integral [[Fids. (a) F(x, y, z) - xi + yj + 2zk, S is the part of the paraboloid z - x2 + y2, 251 (b) F(x, y, z) = (z, x-z, y), S is the triangle with vertices (1,0,0), (0, 1,0), and (0,0,1), oriented downward (c) F-(y. -x,z), S is the upward helicoid parametrized by r(u, v) = (UCOS v, usin v,V), osus 2, OSVS (Hint: Tu x Ty = (sin v, -cos v, u).)...
scalar functions of position, ?(x, y, z) w(x,y.z) be vector functions of position. By writing the subscripted component form. verify the following identities. 5. Let and ?(x,y,z) be and let v(x, y, z) and (b) Div(v +w)- Div v + Div w (c) Div(pv)-(Vp) v+(Div v)
you are given two vectors: v=[x2 +y2+ z2, 2xyz, x+y+2z] u=[xy+z , xy2 z2 , x+3z] Calculate the following expressions: a) v+u b) v•u c) v x u d) div v