Partial Derivatives and Gradients

Functions of two variables use two inputs to determine one output. The output depends on both independent variables at the same time. They are often written as z = f(x, y), where each ordered pair in the domain gives one unique output value.

The domain of a function of two variables is the region in the horizontal plane that contains the input pairs. Each point in this region matches one output value. That output can be shown as a vertical height above the plane. When all of the heights are...

Weather maps show how a function of two variables can describe temperature across a region. In this setting, each place is identified by two coordinates, longitude and latitude. Since one temperature value is assigned to each coordinate pair, the data can be written as z = f(x, y).

The variables x and y represent the horizontal position on the map. The variable z represents temperature. This means that every point (x, y) in the region has a matching temperature value.

A graph of this...

Contour lines make it possible to read the shape of a surface on a flat map. In mathematics, these lines are called level curves. They show points in a domain where a function of two variables has the same output value.

For a function written as z = f(x, y), a level curve is defined by f(x, y) = k, where k is a constant. That constant represents one fixed height or elevation. Along one contour line, the function value stays the same even when the x and y coordinates change.

This idea helps...

A function of three variables gives one real number for each point in three-dimensional space. Each point is written with Cartesian coordinates, x, y, and z, and the function maps that ordered triple to a single value. These functions are often used to describe physical quantities that change across space.

One example is the electric potential created by a point charge. Here, the potential at a location depends only on the distance from the charge. If the charge is placed at the origin, the...

Path independence is the key idea behind limits of multivariable functions. For a function of two variables, the input can approach a point in the plane from many directions. That makes multivariable limits more complex than single-variable limits, where the input moves along one line and can approach from only two directions.

For a function written as z = f(x, y), the notation lim(x,y)→(a,b) f(x,y) = L means the output values move toward L as the point (x, y) gets closer to (a, b). A...

Limit laws in multivariable calculus give clear rules for finding limits of functions with more than one variable. These rules help break a complex expression into smaller parts whose limits are already known. They are used when \lim_{(x,y)\to(a,b)} f(x,y)=L and \lim_{(x,y)\to(a,b)} g(x,y)=M exist.

The sum law says the limit of a sum equals the sum of the limits. The difference law says the limit of a difference equals the difference of the limits. In notation,...

Continuity in multivariable functions describes how a function with two or more input variables changes near a point. It extends the idea of continuity from single-variable calculus into higher dimensions. This idea is important for modeling real-world situations that vary across space.

A multivariable function is continuous at a point only if three conditions are met at the same time. The function must be defined at that point. The limit must exist as the input moves toward the point from any...

Partial derivatives help describe how a multivariable function changes when one input changes and the others stay fixed. In many real-world situations, one output depends on more than one input. A high-tech assembly plant is one example. Total production can depend on technician labor and machine capacity at the same time.

This relationship can be written as a continuous function P(T, M), where T stands for technician labor input and M stands for machine capacity. When demand rises but the...

Partial derivatives describe how a surface changes in one direction at a time. A surface from a function of two variables can be pictured like uneven terrain. Each point on that terrain is located with Cartesian coordinates, and the function gives the height at each pair of horizontal coordinates.

To study the surface at one point, you can move in a single direction while holding the other variable fixed. If you move along the x-direction and keep y constant, the change in height gives the...

Partial derivatives and surface curvature help describe multivariable functions. A multivariable function gives one output for each ordered set of inputs. For a function f(x, y), each point (x, y) can be treated as a height z = f(x, y). This creates a surface in three-dimensional space. These functions often appear in physical models and optimization problems, where behavior depends on several changing variables.

To study how the function changes in one direction, one variable is held fixed...

Partial differential equations help describe waves that change across both space and time. A stone dropped into a still pond creates circular waves that spread outward. The water surface rises and falls at each point as the wave passes, while the shape of the surface also changes across the pond at any fixed moment.

This kind of motion needs a mathematical model that tracks more than one variable at once. At one location, the water level changes as time moves forward. That change includes the...

Partial derivatives help describe a tangent plane for a surface in multivariable calculus. For a surface written as z = f(x, y), the tangent plane at one point gives the best linear approximation near that point. This makes a curved surface easier to study with a simple planar model.

To build the tangent plane, take vertical slices through the surface. Hold x constant in one slice and y constant in another slice. Each slice forms a curve on the surface. The tangent lines to those curves at the...

Tangent plane estimates help approximate a differentiable function of two variables near a known point. Instead of using the curved surface directly, the function is replaced by its tangent plane. This gives a fast estimate for values close to the point being studied.

For the function f(x,y)=x^2+3y^2, the point (2, 1) is used as the reference. The exact value there is f(2,1)=2^2+3(1)^2=4+3=7. The linear approximation near (a,b) is L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b).

To build the...

Total derivatives in connected variables show how one change can affect several related quantities. Here, z depends on two intermediate variables, x and y, and both x and y change with a third variable, t. The connection is indirect, because t does not appear in the formula for z itself.

The goal is to find the total derivative of z with respect to t, written as dz/dt. When the functions are differentiable, the change in z can be split into two parts. One part comes from how z changes with x...

Implicit differentiation with partial derivatives helps find tangent slopes when two variables are linked by one equation. The relationship is not solved for one variable first. Instead, both variables stay connected in the same implicit relation.

To work with this kind of equation, one variable is treated as depending on the other. Then the chain rule is applied to every term in the equation. Because of that setup, the derivative includes partial derivatives and also the derivative of the...

Directional derivatives measure how a surface changes in any chosen direction. In multivariable calculus, partial derivatives show change along one coordinate at a time. For a function of two variables, one partial derivative gives the slope in the x-direction, and the other gives the slope in the y-direction.

These single-direction slopes are useful for local behavior. However, many real movements do not follow the coordinate axes. Fluid flow, heat transfer, and motion across uneven terrain...

The gradient vector shows the direction of steepest ascent on a surface. In this lesson, a topographical map is used to model land as a height surface that depends on horizontal position. Each point on the map has an elevation, so a hiker at one location can move in many directions, but only one path gives the fastest increase in height.

That special path is called the direction of steepest ascent. In multivariable calculus, the gradient vector represents it. The gradient points toward higher...

The directional derivative shows how a multivariable function changes at a point when you move in a chosen direction. That direction is written as a unit vector, so only the direction matters and not the length of the vector. Changing the direction changes the rate of change, which is why the directional derivative depends strongly on the path you choose.

The directional derivative is found using the gradient vector and a unit direction vector. The gradient tells you both the direction and the...

Gradient vectors and tangent planes help describe the shape of a level surface at a point. A level surface is the set of all points in space where a function of three variables has the same fixed value. To understand the local geometry near one point, it is not enough to know the coordinates alone. You also need to know how the surface tilts there.

One way to study that tilt is to follow a path that stays on the level surface and passes through the point. This path can be written as a...

The gradient vector describes how a surface changes and where it rises fastest. For a function of two variables, the surface can be studied by looking at how it changes in specific directions. This makes the gradient useful for measuring local steepness at a point.

If one variable is held constant, the surface becomes a cross-sectional curve. Moving parallel to a coordinate axis shows how the other variable changes. The slope of that curve at a point gives the directional rate of change in...

Multivariable calculus uses critical points to locate local maximum and minimum values on a surface. A local maximum is a point where the function value is higher than at nearby points. A local minimum is a point where the function value is lower than at nearby points. These points are called local extrema, and they matter in optimization problems.

Local extrema occur at critical points, where the surface is momentarily flat in all directions. At a critical point, the partial derivatives are...

Lagrange multipliers help solve constrained optimization problems. In these problems, you maximize or minimize a quantity while keeping a fixed condition. A common example is a rectangular pen built against a barn wall using 100 meters of fencing.

The barn wall makes one side of the pen, so only the other three sides need fencing. The goal is to find the dimensions that give the greatest area. Let L be the length parallel to the wall and W be the width perpendicular to the wall. Then the area...

Lagrange multipliers with two constraints help find a maximum or minimum when a function must satisfy two conditions at once. The goal may be to optimize cost, area, distance, or energy, but only among solutions that meet both requirements. In many problems, the constraints represent fixed volume, limited resources, or prescribed dimensions.

For a function of three variables, each constraint makes a surface in three-dimensional space. A feasible point is a point that satisfies both constraints.

A silo with a cylindrical base and a hemispherical roof can be sized with Lagrange multipliers. This calculus method helps find the best design when the surface area must stay fixed. In this problem, the goal is to maximize storage volume while using a set amount of material.

The model uses two main variables: the radius r of the base and the height h of the cylindrical section. The total volume comes from adding the volume of the cylinder and the hemisphere. The fixed surface area includes...