gradient

Examples of gradient in the following topics:

• Fundamental Theorem for Line Integrals

  • Gradient theorem says that a line integral through a gradient field can be evaluated from the field values at the endpoints of the curve.
  • The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.
  • The gradient theorem implies that line integrals through irrotational vector fields are path-independent.
  • The gradient theorem also has an interesting converse: any conservative vector field can be expressed as the gradient of a scalar field.
  • Electric field is a vector field which can be represented as a gradient of a scalar field, called electric potential.

• Directional Derivatives and the Gradient Vector

  • If the function $f$ is differentiable at $\mathbf x$, then the directional derivative exists along any vector $\mathbf v$, and one has $\nabla_{\mathbf{v}}{f}(\mathbf{x}) = \nabla f(\mathbf{x}) \cdot \mathbf{v}$, where the $\nabla f(\mathbf{x})$ is the gradient vector and $\cdot$ is the dot product.
  • The gradient of the function $f(x,y) = −\left((\cos x)^2 + (\cos y)^2\right)$ depicted as a projected vector field on the bottom plane.

• Conservative Vector Fields

  • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
  • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
  • Here $\nabla\varphi$ denotes the gradient of $\varphi$.
  • If $\mathbf{v}=\nabla\varphi$ is a conservative vector field, then the gradient theorem states that $\int_P \mathbf{v}\cdot d\mathbf{r}=\varphi(B)-\varphi(A)$.

• Vector Fields

  • Gradient field: Vector fields can be constructed out of scalar fields using the gradient operator (denoted by the del: ∇).
  • A vector field V defined on a set S is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that:

• Lagrange Multiplers

  • Since the gradient of a function is perpendicular to the contour lines, this is the same as saying that the gradients of $f$ and $g$ are parallel.
  • $\nabla_{x,y} f = - \lambda \nabla_{x,y} g$, where $\nabla_{x,y} f= \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)$ and $\nabla_{x,y} g= \left( \frac{\partial g}{\partial x}, \frac{\partial g}{\partial y} \right)$ are the respective gradients.
  • The constant is required because, although the two gradient vectors are parallel, the magnitudes of the gradient vectors are generally not equal.
  • Note that $\lambda \neq 0$; otherwise we cannot assert the two gradients are parallel.

• Stokes' Theorem

  • Therefore, electric field can be written as a gradient of a scalar field:
  • As we have seen in our previous atom on gradient theorem, this simply means:

• Curl and Divergence

  • The four most important differential operators are gradient, curl, divergence, and Laplacian.
  • The four most important differential operators are gradient ($\nabla f$), curl ($\nabla \times \mathbf{F}$), divergence ($\nabla \cdot \mathbf{F}$), and Laplacian ($\nabla^2 f = \nabla \cdot \nabla f$) .
  • Gradient, curl, divergence, and Laplacian are four most important differential operators.

• Functions of Several Variables

  • The theorems we learned are gradient theorem, Stokes' theorem, divergence theorem, and Green's theorem.

• Linear Equations

  • In this particular equation, the constant $m$ determines the slope or gradient of that line, and the constant term $b$ determines the point at which the line crosses the $y$-axis, otherwise known as the $y$-intercept.

• Differentiation and Rates of Change in the Natural and Social Sciences

  • Geologists study the rate of earth shift and the temperature gradient of rocks near a volcano.