(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.
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