(a) To define: Curl F Definition: Curl of F : If F x , y , z = P □ i + Q □ j + R □ k is a vector field on R 3 and the partial derivatives of P , Q , R exists, then the curl of F is vector field on R 3 defined by, c u r l F = ∂ R ∂ y - ∂ Q ∂ z i + ∂ P ∂ z - ∂ R ∂ x j + ∂ Q ∂ x - ∂ P ∂ y k = ∇ × F (b) To define: Div F Definition: Divergence of F : If F x , y , z = P □ i + Q □ j + R □ k is a vector field on R 3 and ∂ P ∂ x , ∂ Q ∂ y , ∂ R ∂ z exists, then the divergence of F is d i v F = d i v P , Q , R = ∂ P ∂ x + ∂ Q ∂ y + ∂ R ∂ z = ∇ · F (c) To explain: The physical interpretations of curl F and div F , if F is a velocity field in fluid flow Explanation: At a point in the fluid, the vector curl F aligns with the axis about which the fluid tends to rotate, and its length measures the speed of rotation; div F at a point measuresthe tendency of the fluidto flow away (diverge) from that point.
(a) To define: Curl F Definition: Curl of F : If F x , y , z = P □ i + Q □ j + R □ k is a vector field on R 3 and the partial derivatives of P , Q , R exists, then the curl of F is vector field on R 3 defined by, c u r l F = ∂ R ∂ y - ∂ Q ∂ z i + ∂ P ∂ z - ∂ R ∂ x j + ∂ Q ∂ x - ∂ P ∂ y k = ∇ × F (b) To define: Div F Definition: Divergence of F : If F x , y , z = P □ i + Q □ j + R □ k is a vector field on R 3 and ∂ P ∂ x , ∂ Q ∂ y , ∂ R ∂ z exists, then the divergence of F is d i v F = d i v P , Q , R = ∂ P ∂ x + ∂ Q ∂ y + ∂ R ∂ z = ∇ · F (c) To explain: The physical interpretations of curl F and div F , if F is a velocity field in fluid flow Explanation: At a point in the fluid, the vector curl F aligns with the axis about which the fluid tends to rotate, and its length measures the speed of rotation; div F at a point measuresthe tendency of the fluidto flow away (diverge) from that point.
Solution Summary: The author explains how to define a Curl of F, if the vector field is on R3 and the partial derivatives exist.
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.
Chapter 16.R, Problem 9CC
To determine
(a)
To define:
Curl F
Definition:
Curl of F:
If Fx,y,z=P□i+Q□j+R□k is a vector field on R3 and the partial derivatives of P,Q,R exists, then the curl of F is vector field on R3 defined by,
curlF=∂R∂y-∂Q∂zi+∂P∂z-∂R∂xj+∂Q∂x-∂P∂yk=∇×F
(b)
To define:
Div F
Definition:
Divergence of F:
If Fx,y,z=P□i+Q□j+R□k is a vector field on R3 and ∂P∂x,∂Q∂y,∂R∂z exists, then the divergence of F is
divF=divP,Q,R=∂P∂x+∂Q∂y+∂R∂z=∇·F
(c)
To explain:
The physical interpretations of curl F and div F, if F is a velocity field in fluid flow
Explanation:
At a point in the fluid, the vector curl F aligns with the axis about which the fluid tends to rotate, and its length measures the speed of rotation; div F at a point measuresthe tendency of the fluidto flow away (diverge) from that point.
Explain how to compute the curl of the vector field F = ⟨ƒ, g, h⟩.
A net is dipped in a river. Determine the
flow rate of water across the net if the
velocity vector field for the river is given
by v=(x-y,z+y+7,z2) and the net is
decribed by the equation y=1-x2-z2, y20,
and oriented in the positive y- direction.
(Use symbolic notation and fractions
where needed.)
Which of the following expressions are meaningful (where F is a vector field and f is a function)? Of those that are meaningful, which are automatically zero?
(a) div(∇f )
(b) curl(∇f )
(c) ∇curl(f )
(d) div(curl(F))
(e) curl(div(F))
(f) ∇(div(F))
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