Gradient of a Scalar T (grad T) - Definition, Significance & Solved Examples.
- The gradient of a scalar field provides a vector field that states how the scalar value is changing throughout space – a change that has both a magnitude and direction.
- A gradient is applied to a scalar quantity that is a function of a 3D vector field: position. The gradient measures the direction in which the scalar quantity changes the most, as well as the rate of change with respect to position.
- The physical meaning of the gradient of a scalar is that it represents the steepness of the slope or line. For example, height is a scalar quantity; gradient of the height would be a vector pointing upwards. The length of the vector is proportional to the steepness of the slope.
- A derivative is required that tells us how fast the function varies, if we move a little distance.
- Consider a scalar function T which is a function of space coordinates x, y and z.
- The projection or the component of ∇ T in the direction of a unit vector a_{l} is ∇ T . a_{l} and is called the DIRECTIONAL DERIVATIVE of T along unit vector. Hence dT/dl is the directional derivative of T.
- Hence we also say that, the gradient of a scalar field indicates the direction of greatest change (that is largest derivative) as well as the magnitude of that change, at every point in space.
PROPERTIES OF GRADIENT OF A SCALAR FIELD T:
- If A =∇ T, T is said to be the scalar potential of A.
- The magnitude of ∇ T equals the maximum rate of change in T per unit distance.
- ∇T points in the direction of the maximum rate of change in V.
- ∇T at any point is perpendicular to the constant T surface that passes through that point.
Gradient of a scalar T for Cartesian coordinate system is given as:
Gradient of a scalar T for Cylindrical coordinate system is given as:
Gradient of a scalar T in Spherical coordinate system is given as:
SOLVED EXAMPLES / NUMERICALS:
Q.4 Find the gradient of these scalar fields:
a) U = 4xz_{2} + 3yz
b) H = r^{2}cosθ cosφ SOLUTION/ANSWER
Q.5 If V_{(x, y, z)} = 3x^{2}y –y^{2}z^{2}, find ∇ V and |∇ V| at the point (1, 2, -1). SOLUTION/ANSWER
Q.6 Given φ = xy +yz +xz, find gradient φ at point (1, 2, 3) and the directional derivative of φ. SOLUTION/ANSWER
Q.7 Find V_{(x, y, z)} if grad V = (y^{2} – 2xyz^{3})a_{x} + (3 + 2xy – x^{2}z^{3})a_{y} + (4z^{3} – 3x^{2}yz^{2})a_{z} and V_{(0, 0, 0)} = -2. SOLUTION/ANSWER
Q.8 Find the unit normal vector of the surface x^{2} + y^{2} + z^{2} = 14 at (-1, 3, 2) ? SOLUTION/ANSWER
Q.9 The temperature in an auditorium is given by T = x^{2} + y^{2} – z. A mosquito located at (1, 1, 2) in the auditorium desires to fly in such a direction that it will get warm as soon as possible. In what direction must it fly? SOLUTION/ANSWER
ALSO READ:
- Line , Surface and Volume Intergral.
- Del Operator - Definition and Significance.
- Gradient Of a Scalar (∇ V).
- Numericals / Solved Examples - Gradient Of a Scalar.
- Divergence Of a Vector ( ∇ . A ).
- Numericals / Solved Examples - Divergence Of a Vector.
- Curl Of a Vector ( ∇ x A).
- Laplacian Of a Scalar ( ∇^{2} V).
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