Find the gradient fields of the functions. f(x, y, z)=(x^2+y^2+z^2)^-1 / 2

Name: Problem 1, Chapter 15: University Calculus: Early Transcendentals
Uploaded: 2020-04-27T18:05:52-08:00
Duration: 1 min 58 s
Channel: Vikash Ranjan
Description: Find the gradient fields of the functions. f(x, y, z)=(x^2+y^2+z^2)^-1 / 2

step 1 problem we have f of x y j that is given as x square plus y square plus j squared to the power o

Find the gradient fields of the functions. f(x, y,...

Key Concepts

Gradient

The gradient is a vector operator that indicates both the direction and the rate of fastest increase of a scalar function. In multivariable calculus, it is computed by taking the collection of the function’s partial derivatives with respect to each of its variables, forming a vector field that points in the direction of maximum increase.

Partial Derivatives

Partial derivatives refer to the derivatives of a function with respect to one variable while holding the other variables constant. In the context of finding a gradient, each partial derivative represents the sensitivity of the function to changes along one coordinate axis, and together they provide the components of the gradient vector.

Chain Rule

The chain rule is a fundamental differentiation technique that is used to compute derivatives of composite functions. In multivariable calculus, when a function is composed of other functions—especially when involving inner functions like the sum of squares— the chain rule systematically accounts for the derivative of the inner function when differentiating the overall expression.

Scalar Field

A scalar field is a function that assigns a single scalar value to every point in a space. In many physical situations, scalar fields represent quantities such as temperature or potential energy, and the gradient of a scalar field produces a vector field that describes the spatial variation of that quantity.

Vector Field

A vector field assigns a vector to every point in a given space. When the gradient operator is applied to a scalar field, the resulting vector field indicates, at each point, the direction in which the function increases most rapidly and the rate of that increase.

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