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 = 4xz₂ + 3yz
b) H = r²cosθ cosφ                SOLUTION/ANSWER


Q.5 If V_{(x, y, z)} = 3x²y –y²z², 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 = (y² – 2xyz³)a_x + (3 + 2xy – x²z³)a_y + (4z³ – 3x²yz²)a_z and V_{(0, 0, 0)} = -2.      SOLUTION/ANSWER


Q.8 Find the unit normal vector of the surface x² + y² + z² = 14 at (-1, 3, 2) ?     SOLUTION/ANSWER


Q.9 The temperature in an auditorium is given by T = x² + y² – 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).

- Short Notes/FAQ's

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