Slope vs. Gradient — What's the Difference?

In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m; there is no clear answer to the question why the letter m is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as "y = mx + b" and it can also be found in Todhunter (1888) who wrote it as "y = mx + c".Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line.

A graduated scale or measure of difference.

The inclined surface of a hill or terrain.

In mathematics, the rise over run of a line on a graph.

A surface that lies at an angle to the horizontal.

In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) ∇ f {\displaystyle \nabla f} whose value at a point p {\displaystyle p} is the vector whose components are the partial derivatives of f {\displaystyle f} at p {\displaystyle p} . That is, for f : R n → R {\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} } , its gradient ∇ f : R n → R n {\displaystyle \nabla f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}} is defined at the point p = ( x 1 , … , x n ) {\displaystyle p=(x_{1},\ldots ,x_{n})} in n-dimensional space as the vector: ∇ f ( p ) = [ ∂ f ∂ x 1 ( p ) ⋮ ∂ f ∂ x n ( p ) ] .

The direction or manner of incline.

A rate of inclination; a slope.

An upward or downward slant.

An ascending or descending part; an incline.

A surface of which one end or side is at a higher level than another; a rising or falling surface

(Physics) The rate at which a physical quantity, such as temperature or pressure, changes in response to changes in a given variable, especially distance.

A person from East Asia, especially Vietnam.

(Mathematics) A vector having coordinate components that are the partial derivatives of a function with respect to its variables.

(of a surface or line) be inclined from a horizontal or vertical line; slant up or down

(Biology) A series of progressively increasing or decreasing differences in the growth rate, metabolism, or physiological activity of a cell, organ, or organism.

Move in an idle or aimless manner

A slope or incline.

To diverge from the vertical or horizontal; incline

A rate of inclination or declination of a slope.

To move or walk

The ratio of the rates of change of a dependent variable and an independent variable, the slope of a curve's tangent.

To cause to slope

(science) The rate at which a physical quantity increases or decreases relative to change in a given variable, especially distance.

An inclined line, surface, plane, position, or direction.

A differential operator that maps each point of a scalar field to a vector pointed in the direction of the greatest rate of change of the scalar. Notation for a scalar field φ: ∇φ

A stretch of ground forming a natural or artificial incline

A gradual change in color. A color gradient; gradation.

A deviation from the horizontal.

Moving by steps; walking.

The amount or degree of such deviation.

Rising or descending by regular degrees of inclination.

The rate at which an ordinate of a point of a line on a coordinate plane changes with respect to a change in the abscissa.

Adapted for walking, as the feet of certain birds.

The tangent of the angle of inclination of a line, or the slope of the tangent line for a curve or surface.

Moving by steps; walking; as, gradient automata.

Offensive Slang Used as a disparaging term for a person of East Asian birth or ancestry.

Rising or descending by regular degrees of inclination; as, the gradient line of a railroad.

An area of ground that tends evenly upward or downward.

Adapted for walking, as the feet of certain birds.

The degree to which a surface tends upward or downward.

The rate of regular or graded ascent or descent in a road; grade.

(mathematics) The ratio of the vertical and horizontal distances between two points on a line; zero if the line is horizontal, undefined if it is vertical.

A part of a road which slopes upward or downward; a portion of a way not level; a grade.

(mathematics) The slope of the line tangent to a curve at a given point.

The rate of increase or decrease of a variable magnitude, or the curve which represents it; as, a thermometric gradient.

The angle a roof surface makes with the horizontal, expressed as a ratio of the units of vertical rise to the units of horizontal length (sometimes referred to as run).

The variation of the concentration of a chemical substance in solution through some linear path; also called concentration gradient; - usually measured in concentration units per unit distance. Concentration gradients are created naturally, e.g. by the diffusion of a substance from a point of high concentration toward regions of lower concentration within a body of liquid; in laboratory techniques they may be made artificially.

A person of Chinese or other East Asian descent.

A graded change in the magnitude of some physical quantity or dimension

(intransitive) To tend steadily upward or downward.

The property possessed by a line or surface that departs from the horizontal;

(transitive) To form with a slope; to give an oblique or slanting direction to; to incline or slant.

The measure of steepness or the degree of inclination.

To try to move surreptitiously.

A rate of change with respect to distance.

(military) To hold a rifle at a slope with forearm perpendicular to the body in front holding the butt, the rifle resting on the shoulder.

A transition from one value to another.

(obsolete) Sloping.

(obsolete) slopingly

An oblique direction; a line or direction including from a horizontal line or direction; also, sometimes, an inclination, as of one line or surface to another.

Any ground whose surface forms an angle with the plane of the horizon.

The part of a continent descending toward, and draining to, a particular ocean; as, the Pacific slope.

Sloping.

In a sloping manner.

To form with a slope; to give an oblique or slanting direction to; to direct obliquely; to incline; to slant; as, to slope the ground in a garden; to slope a piece of cloth in cutting a garment.

To take an oblique direction; to be at an angle with the plane of the horizon; to incline; as, the ground slopes.

To depart; to disappear suddenly.

An elevated geological formation;

The property possessed by a line or surface that departs from the horizontal;

Be at an angle;

The degree to which a surface deviates from the horizontal.

Grade.

No, while often used for terrain, "slope" can describe any inclined surface.

As "rise over run" or Δy/Δx.

A surface that rises or falls sharply.

To describe a smooth transition between colors or shades.

It has a zero or no slope.

Often, especially in math, but it can also describe a general transition, like in color gradients.

Yes, particularly in linear equations to describe the incline of a line.

Less commonly, but it's mostly used for tangible inclines.

"Slope" describes the incline, while "elevation" denotes the height above a reference point.

"Gradient," especially when discussing rates of change.

Yes, it can denote any rate of change, like temperature or pressure gradients.

"Gradient" is more specific, often indicating a measured rate of change.

Yes, for example, to describe the steepness of landforms.

No, it can be negative, indicating a decline or decrease.