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In mathematics, the gradient is a generalization of the usual concept of derivative of a function in one dimension to a function in several dimensions. If is a differentiable, scalar-valued function of standard Cartesian coordinates in Euclidean space, its gradient is the vector whose components are the ''n'' partial derivatives of ''f''. It is thus a vector-valued function. Similarly to the usual derivative, the gradient represents the slope of the tangent of the graph of the function. More precisely, the gradient points in the direction of the greatest rate of increase of the function and its magnitude is the slope of the graph in that direction. The components of the gradient in coordinates are the coefficients of the variables in the equation of the tangent space to the graph. This characterizing property of the gradient allows it to be defined independently of a choice of coordinate system, as a vector field whose components in a coordinate system will transform when going from one coordinate system to another. The Jacobian is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds. A further generalization for a function between Banach spaces is the Fréchet derivative. ==Motivation== Consider a room in which the temperature is given by a scalar field, , so at each point the temperature is . (We will assume that the temperature does not change over time.) At each point in the room, the gradient of ''T'' at that point will show the direction the temperature rises most quickly. The magnitude of the gradient will determine how fast the temperature rises in that direction. Consider a surface whose height above sea level at a point (''x'', ''y'') is ''H''(''x'', ''y''). The gradient of ''H'' at a point is a vector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector. The gradient can also be used to measure how a scalar field changes in other directions, rather than just the direction of greatest change, by taking a dot product. Suppose that the steepest slope on a hill is 40%. If a road goes directly up the hill, then the steepest slope on the road will also be 40%. If, instead, the road goes around the hill at an angle, then it will have a shallower slope. For example, if the angle between the road and the uphill direction, projected onto the horizontal plane, is 60°, then the steepest slope along the road will be 20%, which is 40% times the cosine of 60°. This observation can be mathematically stated as follows. If the hill height function ''H'' is differentiable, then the gradient of ''H'' dotted with a unit vector gives the slope of the hill in the direction of the vector. More precisely, when ''H'' is differentiable, the dot product of the gradient of ''H'' with a given unit vector is equal to the directional derivative of ''H'' in the direction of that unit vector. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「gradient」の詳細全文を読む スポンサード リンク
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