Gradient Vector in Multivariable Calculus: Definition & Examples

This comprehensive lesson covers the complete fundamentals of vector algebra. The dot product (scalar product) is defined as A·B = |A| |B| cos(θ). Key properties include: (1) Unit vectors i, j, k have magnitude 1 and are mutually perpendicular, (2) Parallel vectors (θ = 0°) give A·B = |A| |B|, (3) Perpendicular vectors (θ = 90°) give A·B = 0, (4) Opposite vectors (θ = 180°) give A·B = -|A| |B|, (5) A·A = |A|², (6) A·B = 0 when vectors are perpendicular. The magnitude of vector A = a₁i + a₂j + a₃k is |A| = √(a₁² + a₂² + a₃²). Direction cosines are (a₁/|A|, a₂/|A|, a₃/|A|). The unit vector along A is A/|A|. For vectors A and B, the dot product in component form is A·B = a₁b₁ + a₂b₂ + a₃b₃. Vector addition is A + B = (a₁ + b₁)i + (a₂ + b₂)j + (a₃ + b₃)k.

Physical quantities are classified into three types: scalar quantities (magnitude only, no direction, e.g., voltage), vector quantities (magnitude and direction, follow parallelogram law of addition), and tension quantities (magnitude and direction, always follow parallelogram law). Vectors are represented symbolically by showing magnitude and direction separately. Unit vectors have magnitude 1 and indicate direction. The triangle law of vector addition states that if two vectors are represented by two sides of a triangle taken in order, the resultant is the third side taken in opposite order. The parallelogram law states that if two vectors are represented by adjacent sides of a parallelogram, the resultant is the diagonal from the same point. Position vectors connect the origin to a point in space. The resultant vector represents the combined effect of all component vectors. The dot product (scalar product) is defined as a·b = |a||b|cosθ, resulting in a scalar value with properties including commutativity, distributivity, and a·a = |a|². The cross product (vector product) results in a vector perpendicular to both original vectors, with magnitude |a × b| = |a||b|sinθ and direction determined by the right-hand rule.

Any vector can be expressed as the product of its modulus and a unit vector in its direction: v = |v|û. The dot product (scalar product) of two vectors a and b is a·b = |a||b|cosθ, where θ is the angle between them. Key properties include: a·a = |a|², commutative (a·b = b·a), distributive (a·(b+c) = a·b + a·c), and a·0 = 0. If a·b = 0 and neither vector is zero, the vectors are perpendicular. The cross product (vector product) is a×b = |a||b|sinθn, where n is perpendicular to both following the right-hand rule. Key properties include: anti-commutative (a×b = -b×a), distributive (a×(b+c) = a×b + a×c), and a×a = 0. If a×b = 0 and neither vector is zero, the vectors are parallel.

A vector is defined by magnitude (size/length) and direction (orientation in space). Vectors are expressed in component form using unit vectors i, j, k along x, y, z axes. The magnitude of v = (x, y, z) is |v| = √(x² + y² + z²). The unit vector is obtained by dividing the vector by its magnitude. The dot product (scalar product) of two vectors a and b is calculated by multiplying corresponding components and summing: a·b = a₁b₁ + a₂b₂ + a₃b₃. Geometrically, a·b = |a||b|cos(θ), where θ is the angle between vectors. This shows the dot product depends on both magnitudes and the cosine of the angle between them.

Vector algebra is the branch of mathematics dealing with vector quantities that have both magnitude and direction, following the vector law of addition (parallelogram law). Key concepts include: (1) Vectors are represented as magnitude multiplied by a unit vector (direction), with unit vectors along coordinate axes being î, ĵ, and k̂; (2) The dot product (scalar product) of two vectors a and b is defined as a·b = |a||b|cosθ, where θ is the angle between them (0° to 180°), and it represents the projection of one vector onto another; (3) The cross product (vector product) a×b = |a||b|sinθn̂ produces a vector perpendicular to both a and b, following the right-hand rule, with magnitude equal to the area of the parallelogram formed by the vectors; (4) Vectors can be classified as equal (same magnitude and direction), parallel (same direction, different magnitudes), collinear (lie on same line), or coplanar (lie in same plane).

Functions of several variables map multiple input variables to a single output value, where the number of variables determines the dimensionality of the graph: one-variable functions produce curves in 2D space, two-variable functions produce surfaces in 3D space, and three-variable functions produce solids in 4D space; importantly, the graph dimensionality depends on the number of variables in the function definition, not on whether all variables appear explicitly in the equation.

A function of several variables is a rule that assigns to each vector in a subset of R^n (the domain) a unique real number, denoted as f(x₁, x₂, ..., xₙ); these functions can be visualized through their graphs (which require n+1 dimensions, limiting practical visualization to n≤2) or through level curves/surfaces (constant-value sets in the domain), with examples including the Cobb-Douglas production function P(L,K) = 1.01L^0.75K^0.25 and the distance function d(x₁,...,xₙ) = √(x₁²+...+xₙ²).

This section introduces functions of several variables, extending single-variable concepts to multiple inputs. A function of two variables is written as Z = f(X, Y), where Z is the dependent variable and X, Y are independent variables. The domain consists of all allowable input combinations satisfying mathematical restrictions. For f(x, y) = ln(xy), the domain requires xy > 0, meaning points in quadrants I and III excluding axes. When finding domains, consider both denominator restrictions and square root requirements—for f(x, y, z) = x / √(9 - x² - y² - z²), solving 9 - x² - y² - z² > 0 yields x² + y² + z² < 9, representing all points inside a sphere of radius 3. Graphing surfaces in 3D space involves recognizing shapes: f(x, y) = √(4 - x² - y²) represents the upper hemisphere of radius 2 because the square root ensures Z ≥ 0. Linear functions represent planes; to sketch them, find X, Y, and Z intercepts by setting pairs of variables to zero.

Functions of two variables can be visualized as surfaces in three-dimensional space R³, where each (x,y) point maps to a height z = f(x,y). For example, f(x,y) = x² + y² forms a paraboloid (bowl shape), while f(x,y) = sin(x)cos(y) creates periodic wavy surfaces. Level curves (contour lines) defined by f(x,y) = c show points with equal function values, similar to altitude lines on maps—for f(x,y) = x² + y², these form concentric circles; for f(x,y) = x² - y², they form hyperbolas. For three-variable functions, level surfaces f(x,y,z) = c in R³ replace graphs in four-dimensional space—for instance, x² + y² + z² = r² represents spheres, while z - xy = 0 represents hyperbolic surfaces.

A function of several variables extends the concept of real-valued functions by taking multiple independent variables (ordered pairs or n-tuples) as inputs and producing a single real number output; for two variables, the graph is a surface in 3D space (z = f(x,y)), while level curves (contour curves) are projections of horizontal slices onto the XY-plane, and for three variables, level surfaces are analogous projections into 3D space, enabling visualization of complex relationships through these cross-sectional views.

Calculus is the mathematics of rate of change. For a single-variable function, the derivative measures how fast the function changes with respect to the input variable. A steeper graph indicates a larger derivative because a small change in x produces a larger change in f(x). The derivative is the limiting value of the rate of change as the interval shrinks to zero.

Partial derivatives extend the concept of derivative (rate of variation) from single-variable calculus to functions of several variables. When one variable is fixed and the other varies, the function becomes a single-variable function that can be differentiated. For f(x,y) = 9 - x² - y², fixing x=1 gives 8 - y² (a parabola), and fixing y=1 gives 8 - x² (another parabola). These cross-sections are parabolas contained within the paraboloid surface. The partial derivative measures the rate of variation of the function when only one variable changes, analogous to how velocity measures the rate of change of position with respect to time.

This segment establishes the conceptual bridge from single-variable to multi-variable calculus. In single-variable calculus, we analyze functions on a plane with only one direction of movement, where the derivative represents the slope of the tangent line indicating increase or decrease. However, in three-dimensional space, there are infinitely many directions to move, requiring new mathematical tools. A multi-variable function z = f(x, y) creates a surface in 3D space, where each point (a, b) maps to a Z-value. By intersecting this surface with vertical planes parallel to the Z-axis (holding one variable constant), we obtain curves whose tangent lines give us partial derivatives. These represent two special cases of directional derivatives, focusing on movement along the coordinate axes rather than arbitrary directions.

Derivatives describe how quantities change with respect to time. Functions model changing quantities like position, price, or population. The derivative represents the instantaneous rate of change at a specific point, calculated as the limit of the average rate of change as the interval approaches zero. This fundamental concept connects calculus to real-world phenomena where change is continuous and measurable.

A derivative measures the rate of change of one variable with respect to another. It answers questions like: How much does distance change when speed changes? How much does demand change when price increases? How much does expenditure change when income increases? The derivative is denoted by dy/dx or f'(x), representing the instantaneous rate of change between two related variables.