Question
In: Math
Compute the derivative of the given vector field F. Evaluate the line integral of F(x,y,z) = (y+z+yz , x+z+xz , x+y+xy )
Compute the derivative of the given vector field F. Evaluate the line integral of
F(x,y,z) = (y+z+yz , x+z+xz , x+y+xy )
over the path C consisting of line segments joining (1,1,1) to (1,1,2), (1, 1, 2) to (1, 3, 2), and (1, 3, 2) to (4, 3, 2) in 3 different ways, along the given path, along the line from (1,1,1) to (4,3,2), and finally by finding an anti-derivative, f, for F.
Solutions
milcah answered 2 years agoRelated Solutions
The vector field given by E (x,y,z) = (yz – 2x) x + xz y +...
The vector field given by E
(x,y,z) = (yz – 2x)
x + xz y + xy
z may represent an electrostatic field?
Why? If so, finding the potential F a from which E may be
obtained.
Compute the line integral of the vector field F(x, y, z) = ⟨−y, x, z⟩ along...
Compute the line integral of the vector field F(x, y, z) = ⟨−y, x,
z⟩ along the curve which is given by the intersection of the
cylinder x 2 + y 2 = 4 and the plane x + y + z = 2 starting from
the point (2, 0, 0) and ending at the point (0, 2, 0) with the
counterclockwise orientation.
Consider F and C below. F(x, y, z) = yz i + xz j + (xy...
Consider F and C below.
F(x, y, z) = yz i + xz j + (xy + 14z) k
C is the line segment from (3, 0, −1) to (6, 4, 2)
(a) Find a function f such that F =
∇f.
f(x, y, z) =
(b) Use part (a) to evaluate
C
∇f · dr along the given curve C.
Consider F and C below. F(x, y, z) = yz i + xz j + (xy...
Consider F and C below.
F(x, y, z) = yz i + xz j + (xy + 14z) k
C is the line segment from (3, 0, −3) to (5, 5, 1)
(a) Find a function f such that F =
∇f.
f(x, y, z) =
(b) Use part (a) to evaluate
C
∇f · dr along the given curve C.
Consider the scalar functions f(x,y,z)g(x,y,z)=x^2+y^2+z^2, g(x,y,z)=xy+xz+yz, and=h(x,y,z)=√xyz Which of the three vector fields ∇f∇f, ∇g∇g and...
Consider the scalar functions
f(x,y,z)g(x,y,z)=x^2+y^2+z^2,
g(x,y,z)=xy+xz+yz,
and=h(x,y,z)=√xyz
Which of the three vector fields ∇f∇f, ∇g∇g and ∇h∇h are
conservative?
3. a) Consider the vector field F(x, y, z) = (2xy2 z, 2x 2 yz, x2...
3. a) Consider the vector field F(x, y, z) = (2xy2 z, 2x 2 yz,
x2 y 2 ) and the curve r(t) = (sin t,sin t cost, cost) on the
interval [ π 4 , 3π 4 ]. Calculate R C F · dr using the definition
of the line integral. [5] b) Find a function f : R 3 → R so that F
= ∇f. [5] c) Verify your answer from (a) using (b) and the
Fundamental...
Given a scalar fifield φ(x, y, z)=3x2-yz and a vector field F(x, y, z)=3xyz2+2xy3j-x2yzk. Find: (i)F.∇φ....
Given a scalar fifield φ(x, y, z)=3x2-yz and a vector
field F(x, y,
z)=3xyz2+2xy3j-x2yzk.
Find:
(i)F.∇φ.
(ii)F×∇φ.
(iii)∇(∇.F).
Evaluate the line integral ∫_C F⋅dr, where F(x,y,z)=−4sin(x)i+3cos(y)j−4xzk and C is given by the vector function...
Evaluate the line integral ∫_C F⋅dr, where
F(x,y,z)=−4sin(x)i+3cos(y)j−4xzk and C is given by the vector
function r(t)=t^6i−t^5j+t^4k , 0≤t≤1.
Evaluate the surface integral S F · dS for the given vector field F and the...
Evaluate the surface integral
S
F · dS
for the given vector field F and the oriented
surface S. In other words, find the flux of
F across S. For closed surfaces, use the
positive (outward) orientation.
F(x, y, z) = xy i + yz j + zx k
S is the part of the paraboloid
z = 6 − x2 − y2 that lies above the square
0 ≤ x ≤ 1, 0 ≤ y ≤ 1,
and has...
Evaluate the surface integral S F · dS for the given vector field F and...
Evaluate the surface integral
S
F · dS
for the given vector field F and the oriented
surface S. In other words, find the flux of
F across S. For closed surfaces, use the
positive (outward) orientation.
F(x, y, z) = x i − z j + y k
S is the part of the sphere
x2 + y2 + z2 = 1
in the first octant, with orientation toward the origin
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