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The gradient theorem states that if the vector field F is the gradient of some scalar-valued function (i.e., if F is conservative), then F is a path-independent vector field (i.e., the integral of F over some piecewise-differentiable curve is dependent only on end points). This theorem has a powerful converse:
Figure 2: Weight (W), the frictional force (F r), and the normal force (F n) acting on a block.Weight is the product of mass (m) and the acceleration of gravity (g).In the case of an object resting upon a flat table (unlike on an incline as in Figures 1 and 2), the normal force on the object is equal but in opposite direction to the gravitational force applied on the object (or the weight of ...
In Cartesian coordinates, the divergence of a continuously differentiable vector field = + + is the scalar-valued function: = = (, , ) (, , ) = + +.. As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
Illustration of tangential and normal components of a vector to a surface. In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the normal component of the vector.
Given a subset S of R n, a vector field is represented by a vector-valued function V: S → R n in standard Cartesian coordinates (x 1, …, x n).If each component of V is continuous, then V is a continuous vector field.
If ,, are the contravariant basis vectors in a curvilinear coordinate system, with coordinates of points denoted by (,,), then the gradient of the tensor field is given by (see [3] for a proof.) = From this definition we have the following relations for the gradients of a scalar field ϕ {\displaystyle \phi } , a vector field v , and a second ...
A line is normal to γ at γ(t) if it passes through γ(t) and is perpendicular to the tangent vector to γ at γ(t). Let T denote the unit tangent vector to γ and let N denote the unit normal vector. Using a dot to denote the dot product, the generating family for the one-parameter family of normal lines is given by F : I × R 2 → R where
The circle S and the curve C have the common tangent line at P, and therefore the common normal line. Close to P, the distance between the points of the curve C and the circle S in the normal direction decays as the cube or a higher power of the distance to P in the tangential direction.