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What is vorticity vector?
The vorticity vector is DelxV = v/r sin(RV)H1, the Curl of the vector V. The unit vector H1, is perpendicular to the plane formed by the radius vector R and and the vector V.
How does vector calculus apply in fluid mechanics?
The velocity at each point in the fluid is a vector. If the fluid is compressible, the divergence of the velocity vector is nonzero in general. In a vortex the curl is nonzero.
What is curl in mathematical terms?
Curl represents the force of rotation in a 3-D vector field. Generally, the curl vector at a given point is the answer to the question, "What would happen if I stuck something there that could spin but couldn't move?" Unless the curl is zero, it would spin perpendicularly to the curl vector (according to the right-hand rule), and the longer the vector is, the faster. Curl is mathematically defined in a given direction as the limit of "circulation over area", i.e. the line integral of a circle around the point, divided by the area of the circle, with the circle shrinking towards the point. More practically, the actual vector can found by taking the cross product of the gradient operator with the function that defines the field: curl_x = ∂F/∂y - ∂F/∂z curl_y = ∂F/∂z - ∂F/∂x curl_z = ∂F/∂x - ∂F/∂y
What is meant by curl of a vector in maths?
In mathematics, the curl of a vector is the maximum rotation on a vector field, oriented perpendicularly to the certain plane. The curl of a vector is defined by this form: ∇ x F = [i . . . . j . . . . . k] [∂/∂x ∂/∂y ∂/∂z] [P. . . Q. . . .R. . ] ...given that F = <P,Q,R> or Pi + Qj + Rk Perform the cross-product of the terms to obtain: ∇ x F = (∂R/∂y - ∂Q/∂z)i + (∂P/∂z - ∂R/∂x)j + (∂Q/∂x - ∂P/∂y)k
Is a vector necessarily changed if it is rotated through an angle?
Yes. A vector is defined as having magnitude and direction (in reference to a fixed frame). Changing either of these properties redefines the vector.
Is curl of vector function F must perpendicular to every vector function f?
No, the curl of a vector field is a vector field itself and is not required to be perpendicular to every vector field f. The curl is related to the local rotation of the vector field, not its orthogonality to other vector fields.
What is the difference between curl and divergence?
Divergence: rate of spread of vector in free space for non closed path. and Curl: rate of spread of vector in free space for closed path.
What is vorticity vector?
The vorticity vector is DelxV = v/r sin(RV)H1, the Curl of the vector V. The unit vector H1, is perpendicular to the plane formed by the radius vector R and and the vector V.
How does vector calculus apply in fluid mechanics?
The velocity at each point in the fluid is a vector. If the fluid is compressible, the divergence of the velocity vector is nonzero in general. In a vortex the curl is nonzero.
What is curl in mathematical terms?
Curl represents the force of rotation in a 3-D vector field. Generally, the curl vector at a given point is the answer to the question, "What would happen if I stuck something there that could spin but couldn't move?" Unless the curl is zero, it would spin perpendicularly to the curl vector (according to the right-hand rule), and the longer the vector is, the faster. Curl is mathematically defined in a given direction as the limit of "circulation over area", i.e. the line integral of a circle around the point, divided by the area of the circle, with the circle shrinking towards the point. More practically, the actual vector can found by taking the cross product of the gradient operator with the function that defines the field: curl_x = ∂F/∂y - ∂F/∂z curl_y = ∂F/∂z - ∂F/∂x curl_z = ∂F/∂x - ∂F/∂y
What is meant by curl of a vector in maths?
In mathematics, the curl of a vector is the maximum rotation on a vector field, oriented perpendicularly to the certain plane. The curl of a vector is defined by this form: ∇ x F = [i . . . . j . . . . . k] [∂/∂x ∂/∂y ∂/∂z] [P. . . Q. . . .R. . ] ...given that F = <P,Q,R> or Pi + Qj + Rk Perform the cross-product of the terms to obtain: ∇ x F = (∂R/∂y - ∂Q/∂z)i + (∂P/∂z - ∂R/∂x)j + (∂Q/∂x - ∂P/∂y)k
Is the magnitude of a vector the same as the angle formed by the vector?
No, the magnitude of a vector is the length of the vector, while the angle formed by a vector is the direction in which the vector points relative to a reference axis. These are separate properties of a vector that describe different aspects of its characteristics.
What is physical interpretation of vector space?
A vector has two properties: magnitude and direction. The representation of a vector is an arrow. The tip of the arrow points to the direction the vector is acting. The length of the arrow represents the magnitude.
How can a null vector be a vector if there is no direction?
A null vector does not have a direction but still satisfies the properties of a vector, namely having magnitude and following vector addition rules. It is often used to represent the absence of displacement or a zero result in a vector operation.
Is a vector necessarily changed if it is rotated through an angle?
Yes. A vector is defined as having magnitude and direction (in reference to a fixed frame). Changing either of these properties redefines the vector.
Is every irrotational vector field conservative?
Yes, every irrotational vector field is conservative because a vector field being irrotational implies that its curl is zero, which, by one of the fundamental theorems of vector calculus, implies that the vector field is conservative.
What is translation along the vector?
Translation along a vector involves moving an object in a specific direction by a specified distance based on the properties of the vector. This operation involves shifting the object without rotating or changing its orientation, following the direction and magnitude of the vector.
