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What is the physical interpretation of gradient of a scalar field and directional derivative?
Wiki User
∙ 13y ago
The elevation of points on a hill is a scalar 'field'. It can have a different value
at every point, but each one is a scalar value.
Imagine a lumpy bumpy irregular hill, and pick a point to talk about, say,
somewhere on the side of the hill.
At that point, the directional derivative of the elevation is the rate at which
the elevation changes leaving the point in that direction.
It has different values in different directions: If you're looking up the hill, then
the d.d. is positive in that direction; if you're looking down the hill, the d.d. is
negative in that direction. If you're looking along the side of the hill, the d.d.
could be zero, because the elevation doesn't change in that particular direction.
The directional derivative is a vector. The direction is whatever direction you're
talking about, and the magnitude is the rate of change in that direction.
The gradient is the vector that's simply the greatest positive directional derivative
at that point. Its direction is the direction of the steepest rise, and its magnitude
is the rate of rise in that direction.
If your hill is, say, a perfect cone, and you're on the side, then the gradient is the
vector from you straight toward the top, with magnitude equal to the slope of the
side of the cone. Any other vector is a directional derivative, with a smaller slope,
and it isn't the gradient.
Wiki User
∙ 13y ago
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What is the physical interpretation of gradient of a scalar field and directional derivative and its application?
If you think of it as a hill, then the gradient points toward the top of the hill. With the same analogy, directional derivatives would tell the slope of the ground in a direction.
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The gradient, at any point P:(x, y, z), of a scalar point function Φ(x, y, z) is a vector that is normal to that level surface of Φ(x, y, z) that passes through point P. The magnitude of the gradient is equal to the rate of change of Φ (with respect to distance) in the direction of the normal to the level surface at point P. Grad Φ, evaluated at a point P:(x0, y0, z0), is normal to the level surface Φ(x, y, z) = c passing through point P. The constant c is given by c = Φ(x0, y0, z0).
What is the physical interpretation of gradient of a scalar field and directional derivative and what are its applications?
say what
What is the physical interpretation of gradient of a scalar field and directional derivative and its application?
If you think of it as a hill, then the gradient points toward the top of the hill. With the same analogy, directional derivatives would tell the slope of the ground in a direction.
What is the physical significance of gradient define?
The gradient of a curve is the rate of change in the dependent variable relative to the independent variable.
What is the physical application of third- and fourth-order derivatives with respect to distance e g first derivative is slope?
in case of derivative w.r.t time first derivative with a variable x gives velocity second derivative gives acceleration thid derivative gives jerk
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It is the directional response of a plant organ to physical contact with a solid object.
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Give the physical interpretation of the substantial derivative?
The first ∂/∂t term is called V the local derivative. The second ~· ∇ term is called the convective derivative. In steady flows, ∂/∂t =0, and only the convective derivative The substantial derivative has a physical meaning: the rate of change of a quantity (mass, energy, momentum) as experienced by an observer that is moving along with the flow. The observations made by a moving observer are affected by the stationary time-rate-of-change of the property (∂f/∂t), but what is observed also depends on where the observer goes as it floats along with the flow (v · ∇f). If the flow takes the observer into a region where, for example, the local energy is higher, then the observed amount of energy will be higher due to this change in location. The rate of change from the point of view of an observer floating along with a flow appears naturally in the equations of change.
What is the physical name for fourth order derivatives?
We call "jerk" the third order derivative of position with respect to time, that is, the variation of acceleration. Some say that the derivative of jerk with respect to time (the fourth derivative of position with repsect to time) is called "jounce" or "snap".
What is the physical significance of the gradient of a vector field?
it used in our practical life.. for ex. in hills r in mountains
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the gradient of a scalar function of any quantity is defined as a vector field having magnitude equal to the maximum space rate of change of the quantity and having a direction identical with the direction of displacement along which the rate of change is maximum.
