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Reshma
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I am not completely satisfied with Griffith's explanation for the geometrical interpretation of a gradient. Can someone elaborate on the geometrical meaning of the magnitude and the direction of a gradient?
arildno said:What are you looking after?
Hand-waving?
If there are some specific points which you find difficult, focus on them one at a time when posting your questions.
Biology said::zzz: :zzz: :zzz: :zzz: :zzz:
Reshma said:I am not completely satisfied with Griffith's explanation for the geometrical interpretation of a gradient. Can someone elaborate on the geometrical meaning of the magnitude and the direction of a gradient?
Reshma said:I am not completely satisfied with Griffith's explanation for the geometrical interpretation of a gradient. Can someone elaborate on the geometrical meaning of the magnitude and the direction of a gradient?
The geometrical interpretation of gradient is a mathematical concept that represents the rate of change of a function in a particular direction. It can be visualized as a vector that points in the direction of the steepest increase of the function at a specific point.
The gradient can be thought of as a generalization of slope in higher dimensions. In one-dimensional space, slope represents the change in the y-coordinate with respect to the change in the x-coordinate. In multidimensional space, the gradient represents the change in the function with respect to each of its variables.
The gradient is used in optimization problems to find the direction of steepest ascent or descent in a function. By taking the gradient and following its direction, we can reach the maximum or minimum value of the function, depending on the problem at hand.
Yes, the gradient can be both positive and negative. A positive gradient indicates an increase in the function while a negative gradient indicates a decrease. The magnitude of the gradient represents the steepness of the function at a specific point.
The gradient is calculated by taking the partial derivatives of a multivariable function with respect to each of its variables. For example, in a three-dimensional space, the gradient would be represented by a vector with three components, each of which is the partial derivative of the function with respect to one of the variables.