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Vector-valued function

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A vector-valued function, also referred to as a vector function, is a <a href="Function_(mathematics)" title="Function (mathematics)">mathematical function</a> of one or more <a href="Variable_(mathematics)" title="Variable (mathematics)">variables</a> whose <a href="Range_of_a_function" title="Range of a function">range</a> is a set of multidimensional <a href="Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a> or <a href="Infinite-dimensional-vector-valued_function" class="mw-redirect" title="Infinite-dimensional-vector-valued function">infinite-dimensional vectors</a>. The input of a vector-valued function could be a scalar or a vector (that is, the <a href="Dimension" title="Dimension">dimension</a> of the <a href="Domain_of_a_function" title="Domain of a function">domain</a> could be 1 or greater than 1); the dimension of the function's domain has no relation to the dimension of its range.

<div class="mw-heading mw-heading2"><h2 id="Example:_Helix">Example: Helix</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Parametric_curve" class="mw-redirect" title="Parametric curve">Parametric curve</a></div>

A common example of a vector-valued function is one that depends on a single <a href="Real_number" title="Real number">real</a> parameter t, often representing <a href="Time" title="Time">time</a>, producing a <a href="Euclidean_vector" title="Euclidean vector">vector</a> v(t) as the result. In terms of the standard <a href="Unit_vector" title="Unit vector">unit vectors</a> i, j, k of <a href="Cartesian_space" class="mw-redirect" title="Cartesian space">Cartesian 3-space</a>, these specific types of vector-valued functions are given by expressions such as

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (t)=f(t)\mathbf {i} +g(t)\mathbf {j} +h(t)\mathbf {k} }</annotation>

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where f(t), g(t) and h(t) are the coordinate functions of the parameter t, and the domain of this vector-valued function is the <a href="Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> of the domains of the functions f, g, and h. It can also be referred to in a different notation:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (t)=\langle f(t),g(t),h(t)\rangle }</annotation>

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The vector r(t) has its tail at the origin and its head at the coordinates evaluated by the function.

The vector shown in the graph to the right is the evaluation of the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 2\cos t,\,4\sin t,\,t\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \langle 2\cos t,\,4\sin t,\,t\rangle }</annotation>

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</math><img src="./c8cb01ab473143d94aaddff4a07be359989049ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.01ex; height:2.843ex;" alt="{\displaystyle \langle 2\cos t,\,4\sin t,\,t\rangle }" loading="lazy"> near t = 19.5 (between 6π and 6.5π; i.e., somewhat more than 3 rotations). The <a href="Helix" title="Helix">helix</a> is the path traced by the tip of the vector as t increases from zero through 8π.

In 2D, we can analogously speak about vector-valued functions as:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (t)=f(t)\mathbf {i} +g(t)\mathbf {j} }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (t)=\langle f(t),g(t)\rangle }</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Linear_case">Linear case</h2></div>

In the <a href="Linear_map" title="Linear map">linear</a> case the function can be expressed in terms of <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} =A\mathbf {x} ,}</annotation>

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where y is an n × 1 output vector, x is a k × 1 vector of inputs, and A is an n × k matrix of <a href="Parameter" title="Parameter">parameters</a>. Closely related is the affine case (linear up to a <a href="Translation_(geometry)" title="Translation (geometry)">translation</a>) where the function takes the form

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} =A\mathbf {x} +\mathbf {b} ,}</annotation>

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where in addition b'' is an n × 1 vector of parameters.

The linear case arises often, for example in <a href="Multiple_regression" class="mw-redirect" title="Multiple regression">multiple regression</a>, where for instance the n × 1 vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {y}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {y}}}</annotation>

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</math><img src="./3dc8de3d8ea01304329ef9518fad7a6d196c4c01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.509ex;" alt="{\displaystyle {\hat {y}}}" loading="lazy"> of predicted values of a <a href="Dependent_variable" class="mw-redirect" title="Dependent variable">dependent variable</a> is expressed linearly in terms of a k × 1 vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\beta }}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\beta }}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {y} }}=X{\hat {\boldsymbol {\beta }}},}</annotation>

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in which X (playing the role of A in the previous generic form) is an n × k matrix of fixed (empirically based) numbers.

<div class="mw-heading mw-heading2"><h2 id="Parametric_representation_of_a_surface">Parametric representation of a surface</h2></div>

A <a href="Surface_(mathematics)" title="Surface (mathematics)">surface</a> is a 2-dimensional set of points embedded in (most commonly) 3-dimensional space. One way to represent a surface is with <a href="Parametric_equation" title="Parametric equation">parametric equations</a>, in which two parameters s and t determine the three <a href="Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> of any point on the surface:

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<annotation encoding="application/x-tex">{\displaystyle (x,y,z)=(f(s,t),g(s,t),h(s,t))\equiv \mathbf {F} (s,t).}</annotation>

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Here F is a vector-valued function. For a surface embedded in n-dimensional space, one similarly has the representation

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<annotation encoding="application/x-tex">{\displaystyle (x_{1},x_{2},\dots ,x_{n})=(f_{1}(s,t),f_{2}(s,t),\dots ,f_{n}(s,t))\equiv \mathbf {F} (s,t).}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Derivative_of_a_three-dimensional_vector_function">Derivative of a three-dimensional vector function</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Gradient" title="Gradient">Gradient</a></div>

Many vector-valued functions, like <a href="Scalar-valued_function" class="mw-redirect" title="Scalar-valued function">scalar-valued functions</a>, can be <a href="Derivative" title="Derivative">differentiated</a> by simply differentiating the components in the Cartesian coordinate system. Thus, if

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (t)=f(t)\mathbf {i} +g(t)\mathbf {j} +h(t)\mathbf {k} }</annotation>

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is a vector-valued function, then

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {r} }{dt}}=f'(t)\mathbf {i} +g'(t)\mathbf {j} +h'(t)\mathbf {k} .}</annotation>

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The vector derivative admits the following physical interpretation: if r(t) represents the <a href="Position_(vector)" class="mw-redirect" title="Position (vector)">position</a> of a particle, then the derivative is the <a href="Velocity" title="Velocity">velocity</a> of the particle

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} (t)={\frac {d\mathbf {r} }{dt}}.}</annotation>

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Likewise, the derivative of the velocity is the <a href="Acceleration" title="Acceleration">acceleration</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {v} }{dt}}=\mathbf {a} (t).}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Partial_derivative">Partial derivative</h3></div>

The <a href="Partial_derivative" title="Partial derivative">partial derivative</a> of a vector function a with respect to a scalar variable q is defined as<a href="#cite_note-dynon19-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {a} }{\partial q}}=\sum _{i=1}^{n}{\frac {\partial a_{i}}{\partial q}}\mathbf {e} _{i}}</annotation>

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where ai is the scalar component of a in the direction of ei. It is also called the <a href="Direction_cosine#Cartesian_coordinates" title="Direction cosine">direction cosine</a> of a and ei or their <a href="Dot_product" title="Dot product">dot product</a>. The vectors e1, e2, e3 form an <a href="Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> fixed in the <a href="Frame_of_reference" title="Frame of reference">reference frame</a> in which the derivative is being taken.

<div class="mw-heading mw-heading3"><h3 id="Ordinary_derivative">Ordinary derivative</h3></div>

If a is regarded as a vector function of a single scalar variable, such as time t, then the equation above reduces to the first <a href="Derivative" title="Derivative">ordinary time derivative</a> of a with respect to t,<a href="#cite_note-dynon19-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {a} }{dt}}=\sum _{i=1}^{n}{\frac {da_{i}}{dt}}\mathbf {e} _{i}.}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Total_derivative">Total derivative</h3></div>

If the vector a is a function of a number n of scalar variables qr (r = 1, ..., n), and each qr is only a function of time t, then the ordinary derivative of a with respect to t can be expressed, in a form known as the <a href="Total_derivative" title="Total derivative">total derivative</a>, as<a href="#cite_note-dynon19-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {a} }{dt}}=\sum _{r=1}^{n}{\frac {\partial \mathbf {a} }{\partial q_{r}}}{\frac {dq_{r}}{dt}}+{\frac {\partial \mathbf {a} }{\partial t}}.}</annotation>

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Some authors prefer to use capital D to indicate the total derivative operator, as in D/Dt. The total derivative differs from the partial time derivative in that the total derivative accounts for changes in a due to the time variance of the variables qr.

<div class="mw-heading mw-heading3"><h3 id="Reference_frames">Reference frames</h3></div>

Whereas for scalar-valued functions there is only a single possible <a href="Frame_of_reference" title="Frame of reference">reference frame</a>, to take the derivative of a vector-valued function requires the choice of a reference frame (at least when a fixed Cartesian coordinate system is not implied as such). Once a reference frame has been chosen, the derivative of a vector-valued function can be computed using techniques similar to those for computing derivatives of scalar-valued functions. A different choice of reference frame will, in general, produce a different derivative function. The derivative functions in different reference frames have a specific <a href="#Derivative_of_a_vector_function_with_nonfixed_bases">kinematical relationship</a>.

<div class="mw-heading mw-heading3"><h3 id="Derivative_of_a_vector_function_with_nonfixed_bases">Derivative of a vector function with nonfixed bases</h3></div>

The above formulas for the derivative of a vector function rely on the assumption that the <a href="Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> vectors e1, e2, e3 are constant, that is, fixed in the reference frame in which the derivative of a is being taken, and therefore the e1, e2, e3 each has a derivative of identically zero. This often holds true for problems dealing with <a href="Vector_field" title="Vector field">vector fields</a> in a fixed coordinate system, or for simple problems in <a href="Physics" title="Physics">physics</a>. However, many complex problems involve the derivative of a vector function in multiple moving reference frames, which means that the basis vectors will not necessarily be constant. In such a case where the basis vectors e1, e2, e3 are fixed in reference frame E, but not in reference frame N, the more general formula for the <a href="#Ordinary_derivative">ordinary time derivative</a> of a vector in reference frame N is<a href="#cite_note-dynon19-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {{}^{\mathrm {N} }d\mathbf {a} }{dt}}=\sum _{i=1}^{3}{\frac {da_{i}}{dt}}\mathbf {e} _{i}+\sum _{i=1}^{3}a_{i}{\frac {{}^{\mathrm {N} }d\mathbf {e} _{i}}{dt}}}</annotation>

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where the superscript N to the left of the derivative operator indicates the reference frame in which the derivative is taken. <a href="#Ordinary_derivative">As shown previously</a>, the first term on the right hand side is equal to the derivative of a in the reference frame where e1, e2, e3 are constant, reference frame E. It also can be shown that the second term on the right hand side is equal to the relative <a href="Angular_velocity" title="Angular velocity">angular velocity</a> of the two reference frames <a href="#Cross_product">cross multiplied</a> with the vector a itself.<a href="#cite_note-dynon19-1">[1]</a> Thus, after substitution, the formula relating the derivative of a vector function in two reference frames is<a href="#cite_note-dynon19-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {{}^{\mathrm {N} }d\mathbf {a} }{dt}}={\frac {{}^{\mathrm {E} }d\mathbf {a} }{dt}}+{}^{\mathrm {N} }\mathbf {\omega } ^{\mathrm {E} }\times \mathbf {a} }</annotation>

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where NωE is the <a href="Angular_velocity" title="Angular velocity">angular velocity</a> of the reference frame E relative to the reference frame N.

One common example where this formula is used is to find the velocity of a space-borne object, such as a <a href="Rocket" title="Rocket">rocket</a>, in the <a href="Inertial_reference_frame" class="mw-redirect" title="Inertial reference frame">inertial reference frame</a> using measurements of the rocket's velocity relative to the ground. The velocity NvR in inertial reference frame N of a rocket R located at position rR can be found using the formula

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<annotation encoding="application/x-tex">{\displaystyle {\frac {{}^{\mathrm {N} }d}{dt}}(\mathbf {r} ^{\mathrm {R} })={\frac {{}^{\mathrm {E} }d}{dt}}(\mathbf {r} ^{\mathrm {R} })+{}^{\mathrm {N} }\mathbf {\omega } ^{\mathrm {E} }\times \mathbf {r} ^{\mathrm {R} }.}</annotation>

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where NωE is the <a href="Angular_velocity" title="Angular velocity">angular velocity</a> of the Earth relative to the inertial frame N. Since velocity is the derivative of position, NvR and EvR are the derivatives of rR in reference frames N and E, respectively. By substitution,

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<annotation encoding="application/x-tex">{\displaystyle {}^{\mathrm {N} }\mathbf {v} ^{\mathrm {R} }={}^{\mathrm {E} }\mathbf {v} ^{\mathrm {R} }+{}^{\mathrm {N} }\mathbf {\omega } ^{\mathrm {E} }\times \mathbf {r} ^{\mathrm {R} }}</annotation>

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where EvR is the velocity vector of the rocket as measured from a reference frame E that is fixed to the Earth.

<div class="mw-heading mw-heading3"><h3 id="Derivative_and_vector_multiplication">Derivative and vector multiplication</h3></div>

The derivative of a product of vector functions behaves similarly to the <a href="Product_rule" title="Product rule">derivative of a product</a> of scalar functions.<a href="#cite_note-2">[a]</a> Specifically, in the case of <a href="#Scalar_multiplication">scalar multiplication</a> of a vector, if p is a scalar variable function of q,<a href="#cite_note-dynon19-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial q}}(p\mathbf {a} )={\frac {\partial p}{\partial q}}\mathbf {a} +p{\frac {\partial \mathbf {a} }{\partial q}}.}</annotation>

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In the case of <a href="#Dot_product">dot multiplication</a>, for two vectors a and b that are both functions of q,<a href="#cite_note-dynon19-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial q}}(\mathbf {a} \cdot \mathbf {b} )={\frac {\partial \mathbf {a} }{\partial q}}\cdot \mathbf {b} +\mathbf {a} \cdot {\frac {\partial \mathbf {b} }{\partial q}}.}</annotation>

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Similarly, the derivative of the <a href="#Cross_product">cross product</a> of two vector functions is<a href="#cite_note-dynon19-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial q}}(\mathbf {a} \times \mathbf {b} )={\frac {\partial \mathbf {a} }{\partial q}}\times \mathbf {b} +\mathbf {a} \times {\frac {\partial \mathbf {b} }{\partial q}}.}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Derivative_of_an_n-dimensional_vector_function">Derivative of an n-dimensional vector function</h3></div>

A function f of a real number t with values in the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} (t)=(f_{1}(t),f_{2}(t),\ldots ,f_{n}(t))}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} (t)=(f_{1}(t),f_{2}(t),\ldots ,f_{n}(t))}</annotation>

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</math><img src="./586448833763531c27d05a543f9db8a4daa5120f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.513ex; height:2.843ex;" alt="{\displaystyle \mathbf {f} (t)=(f_{1}(t),f_{2}(t),\ldots ,f_{n}(t))}" loading="lazy">. Its derivative equals

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} '(t)=(f_{1}'(t),f_{2}'(t),\ldots ,f_{n}'(t)).}</annotation>

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If f is a function of several variables, say of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \mathbb {R} ^{m}}">

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<annotation encoding="application/x-tex">{\displaystyle t\in \mathbb {R} ^{m}}</annotation>

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</math><img src="./acdbadbbb68944c5b83efe32499399abff917285.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.033ex; height:2.343ex;" alt="{\displaystyle t\in \mathbb {R} ^{m}}" loading="lazy">, then the partial derivatives of the components of f form a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times m}">

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<annotation encoding="application/x-tex">{\displaystyle n\times m}</annotation>

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</math><img src="./d82325a2a02ad79bc7c347ba9702ad46eb0de824.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle n\times m}" loading="lazy"> matrix called the <a href="Jacobian_matrix" class="mw-redirect" title="Jacobian matrix">Jacobian matrix</a> of f.

<div class="mw-heading mw-heading2"><h2 id="Infinite-dimensional_vector_functions">Infinite-dimensional vector functions</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Infinite-dimensional-vector_function" class="mw-redirect" title="Infinite-dimensional-vector function">Infinite-dimensional-vector function</a></div>

If the values of a function f lie in an <a href="Dimension_(vector_space)" title="Dimension (vector space)">infinite-dimensional</a> <a href="Vector_space" title="Vector space">vector space</a> X, such as a <a href="Hilbert_space" title="Hilbert space">Hilbert space</a>, then f may be called an infinite-dimensional vector function.

<div class="mw-heading mw-heading3"><h3 id="Functions_with_values_in_a_Hilbert_space">Functions with values in a Hilbert space</h3></div>

If the <a href="Argument_of_a_function" title="Argument of a function">argument</a> of f is a real number and X is a Hilbert space, then the derivative of f at a point t can be defined as in the finite-dimensional case:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} '(t)=\lim _{h\to 0}{\frac {\mathbf {f} (t+h)-\mathbf {f} (t)}{h}}.}</annotation>

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Most results of the finite-dimensional case also hold in the infinite-dimensional case too, <a href="Mutatis_mutandis" title="Mutatis mutandis">mutatis mutandis</a>. Differentiation can also be defined to functions of several variables (e.g., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle t\in \mathbb {R} ^{n}}</annotation>

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</math><img src="./e2f17bda90bdd7d98cf449af6fd88fc67ae7b0d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.577ex; height:2.343ex;" alt="{\displaystyle t\in \mathbb {R} ^{n}}" loading="lazy"> or even <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in Y}">

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<annotation encoding="application/x-tex">{\displaystyle t\in Y}</annotation>

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</math><img src="./b8e6d32da5df08c8e257e28bfc55577ebd4b82cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.454ex; height:2.176ex;" alt="{\displaystyle t\in Y}" loading="lazy">, where Y is an infinite-dimensional vector space).

N.B. If X is a Hilbert space, then one can easily show that any derivative (and any other <a href="Limit_(mathematics)" title="Limit (mathematics)">limit</a>) can be computed componentwise: if

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} =(f_{1},f_{2},f_{3},\ldots )}</annotation>

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(i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} =f_{1}\mathbf {e} _{1}+f_{2}\mathbf {e} _{2}+f_{3}\mathbf {e} _{3}+\cdots }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} =f_{1}\mathbf {e} _{1}+f_{2}\mathbf {e} _{2}+f_{3}\mathbf {e} _{3}+\cdots }</annotation>

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</math><img src="./bc4827eb003c6e9991e488a6d919f8fac3af1df7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.815ex; height:2.509ex;" alt="{\displaystyle \mathbf {f} =f_{1}\mathbf {e} _{1}+f_{2}\mathbf {e} _{2}+f_{3}\mathbf {e} _{3}+\cdots }" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3},\ldots }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3},\ldots }</annotation>

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</math><img src="./495c8d2b0662d87eeee58210593c688f654b297a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.663ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3},\ldots }" loading="lazy"> is an <a href="Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> of the space X ), and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(t)}">

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<annotation encoding="application/x-tex">{\displaystyle f'(t)}</annotation>

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</math><img src="./c2b503f27f6df6a6fbc77596c078bb6a25311557.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.654ex; height:3.009ex;" alt="{\displaystyle f'(t)}" loading="lazy"> exists, then

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} '(t)=(f_{1}'(t),f_{2}'(t),f_{3}'(t),\ldots ).}</annotation>

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However, the existence of a componentwise derivative does not guarantee the existence of a derivative, as componentwise convergence in a Hilbert space does not guarantee convergence with respect to the actual <a href="Topological_space" title="Topological space">topology</a> of the Hilbert space.

<div class="mw-heading mw-heading3"><h3 id="Other_infinite-dimensional_vector_spaces">Other infinite-dimensional vector spaces</h3></div>

Most of the above hold for other <a href="Topological_vector_space" title="Topological vector space">topological vector spaces</a> X too. However, not as many classical results hold in the <a href="Banach_space" title="Banach space">Banach space</a> setting, e.g., an <a href="Absolutely_continuous" class="mw-redirect" title="Absolutely continuous">absolutely continuous</a> function with values in a <a href="Radon%E2%80%93Nikodym_property" class="mw-redirect" title="Radon–Nikodym property">suitable Banach space</a> need not have a derivative anywhere. Moreover, in most Banach spaces setting there are no orthonormal bases.

<div class="mw-heading mw-heading2"><h2 id="Vector_field">Vector field</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="Vector_field" title="Vector field">Vector field</a>.[<a class="external text external" href="https://en.wikipedia.org/w/index.php?title=Vector_field&action=edit">edit</a>]</div><div class="excerpt">

In <a href="Vector_calculus" title="Vector calculus">vector calculus</a> and <a href="Physics" title="Physics">physics</a>, a <a href="Vector_field" title="Vector field">vector field</a> is an assignment of a <a href="Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vector</a> to each point in a <a href="Space_(mathematics)" title="Space (mathematics)">space</a>, most commonly <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy">.<a href="#cite_note-Vector_field_Galbis-2012-p12-3">[2]</a> A vector field on a <a href="Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a> can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout <a href="Three_dimensional_space" class="mw-redirect" title="Three dimensional space">three dimensional space</a>, such as the <a href="Wind" title="Wind">wind</a>, or the strength and direction of some <a href="Force" title="Force">force</a>, such as the <a href="Magnetic_field" title="Magnetic field">magnetic</a> or <a href="Gravity" title="Gravity">gravitational</a> force, as it changes from one point to another point.

The elements of <a href="Differential_and_integral_calculus" class="mw-redirect" title="Differential and integral calculus">differential and integral calculus</a> extend naturally to vector fields. When a vector field represents <a href="Force" title="Force">force</a>, the <a href="Line_integral" title="Line integral">line integral</a> of a vector field represents the <a href="Work_(physics)" title="Work (physics)">work</a> done by a force moving along a path, and under this interpretation <a href="Conservation_of_energy" title="Conservation of energy">conservation of energy</a> is exhibited as a special case of the <a href="Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a>. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the <a href="Divergence" title="Divergence">divergence</a> (which represents the rate of change of <a href="Volume" title="Volume">volume</a> of a flow) and <a href="Curl_(mathematics)" title="Curl (mathematics)">curl</a> (which represents the rotation of a flow).

A vector field is a special case of a vector-valued function, whose domain's dimension has no relation to the dimension of its range; for example, the <a href="Position_vector" class="mw-redirect" title="Position vector">position vector</a> of a <a href="Space_curve" class="mw-redirect" title="Space curve">space curve</a> is defined only for smaller subset of the ambient space.

Likewise, n <a href="Coordinate_system" title="Coordinate system">coordinates</a>, a vector field on a domain in n-dimensional Euclidean space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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Vector fields are often discussed on <a href="Open_set" title="Open set">open subsets</a> of Euclidean space, but also make sense on other subsets such as <a href="Surface_(topology)" title="Surface (topology)">surfaces</a>, where they associate an arrow tangent to the surface at each point (a <a href="Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">tangent vector</a>).

More generally, vector fields are defined on <a href="Differentiable_manifold" title="Differentiable manifold">differentiable manifolds</a>, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a <a href="Section_(fiber_bundle)" title="Section (fiber bundle)">section</a> of the <a href="Tangent_bundle" title="Tangent bundle">tangent bundle</a> to the manifold). Vector fields are one kind of <a href="Tensor_field" title="Tensor field">tensor field</a>.</div></div>

<ul><li><a href="Coordinate_vector" title="Coordinate vector">Coordinate vector</a></li>

<li><a href="Curve" title="Curve">Curve</a></li>

<li><a href="Multivalued_function" title="Multivalued function">Multivalued function</a></li>

<li><a href="Parametric_surface" title="Parametric surface">Parametric surface</a></li>

<li><a href="Position_vector" class="mw-redirect" title="Position vector">Position vector</a></li>

<li><a href="Parametrization_(geometry)" title="Parametrization (geometry)">Parametrization</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> In fact, these relations are derived applying the <a href="Product_rule" title="Product rule">product rule</a> componentwise.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFKaneLevinson1996" class="citation book cs1">Kane, Thomas R.; Levinson, David A. (1996). "1–9 Differentiation of Vector Functions". Dynamics: Theory and Applications. Sunnyvale, California: McGraw-Hill. pp. 29–37.</cite>

</li>

<li id="cite_note-Vector_field_Galbis-2012-p12-3"><a href="#cite_ref-Vector_field_Galbis-2012-p12_3-0">^</a> <cite id="CITEREFGalbis,_AntonioMaestre,_Manuel2012" class="citation book cs1">Galbis, Antonio; Maestre, Manuel (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tdF8uTn2cnMC&pg=PA12">Vector Analysis Versus Vector Calculus</a>. Springer. p. 12. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4614-2199-3</bdi>.</cite>

</li>

</ol></div></div>

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<ul><li><cite id="CITEREFHuYang2013" class="citation book cs1">Hu, Chuang-Gan; Yang, Chung-Chun (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yuP6CAAAQBAJ">Vector-Valued Functions and their Applications</a>. Springer Science & Business Media. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-94-015-8030-4</bdi>.</cite></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://ltcconline.net/greenl/courses/202/vectorFunctions/vectorFunctions.htm">Vector-valued functions and their properties (from Lake Tahoe Community College)</a></li>

<li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/VectorFunction.html">"Vector Function"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li>

<li><a rel="nofollow" class="external text" href="http://www.everything2.com/index.pl?node_id=1525585">Everything2 article</a></li>

<li><a rel="nofollow" class="external text" href="https://math.etsu.edu/MultiCalc/Chap1/Chap1-6/part1.htm">3 Dimensional vector-valued functions (from East Tennessee State University)</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20050825140840/http://math.etsu.edu/MultiCalc/Chap1/Chap1-6/part1.htm">Archived</a> 2005-08-25 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li>

<li><a rel="nofollow" class="external text" href="http://www.khanacademy.org/video/position-vector-valued-functions?playlist=Calculus">"Position Vector Valued Functions"</a> <a href="Khan_Academy" title="Khan Academy">Khan Academy</a> module</li></ul>

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