- 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).
Your suggestions and comments are welcome in this section. If you want to share something or if you have some stuff of your own, please do post them in the comments section.
anu un?
ReplyDelete