Gradient of a Scalar Field | Engineering Physics

Transcript

what is gradient of a scalar field consider air on the Earth's surface we know that density is a scalar quantity we also know that density of air is same at equal height from the Earth's surface but as we move away from the Earth's surface density of air decreases since density depends on its position in space therefore it can be represented as a function of three axes x y and Zed now consider any point say a in space having a density D if we take the partial derivative of D at Point a with respect to the axis x y and Zed and then add them using the vector sum we get a vector quantity the magnitude of this vector quantity indicates the maximum rate at which the density increases whereas the direction of this Vector indicates the direction at which the density increases the most such a vector quantity is known as the gradient of density of air thus gradient of a scalar field is a vector Vector that represents both the magnitude and direction of the maximum space rate of increase of a scalar field mathematically gradient of a scalar field is expressed as where D is the vector operator it can also be expressed as or note that gradient of a scalar field at any point is perpendicular to the tangent drawn at the surface where the scalar field remains constant the related terms are